Bounds on extremal functions of forbidden patterns

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Bounds on extremal functions of forbidden patterns

by

Jesse Geneson

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2015

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Massachusetts Institute of Technology 2015. All rights reserved.

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Department of Mathematics

May 10, 2015

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Certified by...

Peter Shor

Professor

Thesis Supervisor

Signature redacted

Accepted by ...

Michel Goemans

Chairman, Department Committee on Graduate Theses

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Bounds on extremal functions of forbidden patterns

by

Jesse Geneson

Submitted to the Department of Mathematics on May 10, 2015, in partial fulfillment of the

requirements for the degree of Doctor of Philosophy

Abstract

Extremal functions of forbidden sequences and 0 - 1 matrices have applications to

many problems in discrete geometry and enumerative combinatorics. We present a new computational method for deriving upper bounds on extremal functions of forbidden sequences. Then we use this method to prove tight bounds on the extremal functions of sequences of the form (12 ... 1)' for 1 > 2 and t > 1, abc(acb)t for t > 0,

and avav'a, such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v. We also prove the existence of infinitely many forbidden 0 - 1 matrices P with non-linear extremal functions for which every strict submatrix of P has a linear extremal function. Then we show that for every d-dimensional permutation matrix

P with k ones, the maximum number of ones in a d-dimensional matrix of sidelength

n that avoids P is 20(k) d-Thesis Supervisor: Peter Shor Title: Professor

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Acknowledgments

Thanks to Peter Shor for advising my thesis research and for improving the bounds on

S2(m) in Lemma 11. Thanks also to Jacob Fox for research advice and for help proving

lower bounds on extremal functions of multidimensional permutation matrices. I also thank Henry Cohn for comments to improve the clarity of the 0 - 1 matrix proofs and for ideas about formation width. Also I thank Peter Shor, Jacob Fox, and Henry Cohn for being on my thesis committee. In addition, I thank Joe Gallian for introducing me to extremal functions of 0 - 1 matrices.

Thanks to Peter Tian for collaborating on the multidimensional 0 - 1 matrix

results and on the Python code for computing formation width. Thanks also to Lilly Shen for collaborating on results about extremal functions of 2-dimensional 0 - 1 matrices, as well as Rohil Prasad and Jonathan Tidor for collaborating on results about formation width. Moreover, I thank Tanya Khovanova for advice on several research projects and on my thesis defense, and for collaborating on research about visibility graphs.

Thanks to PRIMES and RSI for giving me the opportunity to collaborate on projects and co-author papers as a research mentor. Thanks also to the NSF and MIT for financial support during graduate school. I also thank Mary Thornton, David Geneson and Arianna Geneson for their support and advice. In addition, I thank Katherine Bian for support, advice, and help on my thesis defense.

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Chapter 1 Introduction

Problems in the area of pattern containment and avoidance focus on determining whether specific substructures are present in a given structure [1, 5, 15, 27, 29, 34,

36, 43, 441. For example, does the sequence a have a subsequence that is isomorphic

to v? Or, analogously, does the 0 - 1 matrix A have a submatrix that can be turned into B by possibly changing some ones to zeroes?

We focus mainly on bounding extremal functions of sequences and 0 - 1 matrices that avoid forbidden patterns. For example, what is the maximum number of ones in an n x n 0 - 1 matrix that does not contain a forbidden pattern? Or, what is the

minimum length k so that every sequence of length k with n distinct letters and no adjacent same letters contains an alternation of length 5?

1.1 Matrix extremal functions

An early motivation for bounding matrix extremal functions was to use them for solving problems in computational and discrete geometry [4, 18, 371. Mitchell wrote an algorithm to find a shortest rectilinear path that avoids obstacles in the plane [371 and proved that the complexity of this algorithm is bounded from above in terms of a specific matrix extremal function, which was bounded by Bienstock and Gybri [4]. Firedi [181 used matrix extremal functions to derive an upper bound on Erd6s and Moser's [101 problem of maximizing the number of unit distances in a convex n-gon. Recent interest in the extremal theory of matrices has been spurred by the resolution of the Stanley-Wilf conjecture using the linearity of the extremal functions of forbidden permutation matrices [33, 36].

The 0 - 1 matrix A contains a 0 - 1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M, then

A avoids M. Let ex(n, M) be the maximum number of ones in an n x n 0 - 1 matrix

that avoids M, and let exk(M, M) be the maximum number of columns in a 0 - 1 matrix with m rows that avoids M and has at least k ones in every column.

Furedi and Hajnal asked for a characterization of all 0 - 1 matrices P such that ex(n, P) = O(n) [19]. A corresponding problem for the column extremal function

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exk(m, P) = 0(m). Another related problem is to characterize all 0 - 1 matrices P

for which eXk(m, P) = OF) ).

A method for bounding ex(n, M) by using bounds on the maximum number of

edges in bar visibility graphs was introduced in [171. By using a similar method with bar visibility hypergraphs, we obtain linear bounds on the extremal functions of other forbidden 0 - 1 matrices.

Call N a minimal nonlinear 0 - 1 matrix if ex(n, N) = w(n) and ex(n, N') = O(n) for every N' properly contained in N. Call the k x 2k matrix P a double permutation matrix if P can be obtained from a k x k permutation matrix by replacing every column with two copies of itself. We prove that ex(n, P) = 2O(k)n for every k x 2k double permutation matrix P. Using this result, we show the existence of infinitely many minimal nonlinear 0 - 1 matrices.

We also study the extremal functions of multidimensional 0 - 1 matrices. A

d-dimensional ni x ... x nd matrix is denoted by A = (a .... ,id), where 1 < i1 < ne for e = 1, 2,. .. , d. We may view a d-dimensional 0 - 1 matrix A (ail ..,id) as a

d-dimensional rectangular box of lattice points with coordinates (i1 , ... , id).

An f-cross section of matrix A is a maximal set of entries ail. ,id with i fixed. A

d-dimensional k x ... x k 0 - 1 matrix is a permutation matrix if each of its i-cross

sections contains a single one for every e = 1,... , d.

A d-dimensional 0 - 1 matrix A avoids another d-dimensional 0 - 1 matrix P if no submatrix of A can be transformed into P by changing some ones to zeroes. The maximum number of ones in a d-dimensional n x ... x n matrix that avoids P is

denoted by f(n, P, d).

We exhibit a family of k x ... x k permutation matrices P for which f(3,d) ₄ has a lower bound of 2Q(kl/d) for n > 2Lkl/dJ/20 Furthermore we improve the upper bound

on A"P") from 20(klogk) to 20(k) for all k x - x k permutation matrices P, and we

show for every fixed d > 2 that the new upper bound is also true for d-dimensional double permutation matrices of dimensions 2k x k x ... x k.

1.2 Sequence extremal functions

A sequence s contains a sequence tt if some subsequence of s can be changed into u by a one-to-one renaming of its letters. If s does not contain u, then s avoids u. The

sequence s is called r-sparse if any r consecutive letters in s are pairwise different.

A Davenport-Schinzel sequence of order s is a 2-sparse sequence that avoids

alter-nations of length s

+

2. Upper bounds on the lengths of Davenport-Schinzel sequences provide bounds on the complexity of lower envelopes of solution sets to linear homo-geneous differential equations of limited order [8] and on the complexity of faces in arrangements of arcs with a limited number of crossings

[1.

A generalized Davenport-Schinzel sequence is an r-sparse sequence that avoids a

fixed forbidden sequence with r distinct letters. Fox et al. [161 and Suk et al. [451 used bounds on the lengths of generalized Davenport-Schinzel sequences to prove that k-quasiplanar graphs on n vertices with no pair of edges intersecting in more than t

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points have at most (n log n)2a("-)' edges, where a(n) denotes the inverse Ackermann function and c is a constant that depends only on k and t.

Our main contribution for sequence extremal functions is a new computational method for proving tight upper bounds using upper bounds that are already known. We also construct families of sequences avoiding alternations in order to prove lower bounds on extremal functions of forbidden alternations.

Let As,k(m) be the maximum number of distinct letters in any sequence which can be partitioned into m contiguous blocks of pairwise distinct letters, has at least

k occurrences of every letter, and avoids alternations of length s. Nivasch [381 proved

that A₅,2d+l(m) = O(mad(m)) for all fixed d > 2. We show that A,+,8(m) =

("F1)

for all s > 2, A₅,₆(M) = 0(mlog log m), and A5,2d+2(M) = O(mad(M)) for all fixed

d > 3.

An (r, s)-formation is a concatenation of s permutations of r letters. If u is a sequence with r distinct letters, then let Ex(u, n) be the maximum length of any r-sparse sequence with n distinct letters which avoids u. We introduce a computational method for deriving tight upper bounds on Ex(u, n): For every sequence u define

fw(u), the formation width of u, to be the minimum s for which there exists r such

that there is a subsequence isomorphic to u in every (r, s)-formation. We use fw(u) to prove upper bounds on Ex(u, n) for sequences a such that a contains an alternation with the same formation width as u.

We generalize the bounds on Ex((ab)t, n) by showing that fw((12. .)t) = 2t - 1

and Ex((12... l)', n) = n2(t2)!a(n)-2O(a(n)t 3

) for every I > 2 and t > 3, such that a(n) denotes the inverse Ackermann function. Upper bounds on Ex((12 ... 1)', n) were

used to bound the maximum number of edges in k-quasiplanar graphs on n vertices with no pair of edges intersecting in more than 0(1) points.

If u is any sequence of the form avav'a such that a is a letter, v is a nonempty

sequence excluding a with no repeated letters and v' is obtained from v by only moving the first letter of v to another place in v, then we show that fw(u) = 4 and Ex(u, n) = ®(na(n)). Furthermore we prove that fw(abc(acb)t) = 2t + 1 and Ex(abc(acb)t, n) = n2 I)(n) O((f)) for every t > 2.

1.3 Order of results

The next two chapters are about sequences. In Chapter 2, we prove bounds on As,k(m), as well as corollaries related to extremal functions of interval chains and

0 - 1 matrices. In Chapter 3, we use formation width to prove tight bounds on

extremal functions of forbidden sequences.

The final two chapters are about 0-1 matrices. In Chapter 4, we show that double permutation matrices and classes of 0 - 1 matrices corresponding to bar visibility hypergraphs have linear extremal functions. In Chapter 5, we generalize Fox's results on permutation matrices, as well as our results on double permutation matrices, to d dimensions.

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Chapter 2 The maximum number of distinct

letters in ababa-free sequences

The sequence s is called r-sparse if any r consecutive letters in s are pairwise dif-ferent. Let D.(n) be the maximum length of any 2-sparse sequence with n distinct letters which avoids alternations of length s. Nivasch [38] and Klazar [32] proved that

lim D5(n) 2, such that a(n) denotes the inverse Ackermann function. Agarwal,

Sharir, Shor [21 and Nivasch [38] proved the bounds D8(n) = n2t!(n)O(o(n)t-1) for

even s > 6 with t == ". Pettie [42] derived sharp bounds on D (n) for all odd s. To define the Ackermann hierarchy let A1(n) = 2n and for k > 2, Ak(0) = 1 and Ak(n) = Akl(Ak(n - 1)) for n > 1. To define the inverse functions let ak(x) =

min {n Ak(n) >

4} for all k > 1.

We define the Ackermann function A(n) to be A,,(3) as in [38]. The inverse Ackermann function a(n) is defined to be min {x : A(x) > n}.

Collections of contiguous distinct letters in a sequence are called blocks. Nivasch's bounds on D8(n) were derived using an extremal function which maximizes number of distinct letters instead of length. Let A.,k(m) be the maximum number of distinct letters in any sequence on m blocks avoiding alternations of length s such that every letter occurs at least k times. Clearly A,(m) = 0 if m < k and A8,k(M) = o0 if k < s - 1 and k < m-.

Nivasch proved that A5,2d+1(mn) = O(Mad(m)) for each fixed d > 2 (but noted

that the bounds on A5,k(m) were not tight for even k). Sundar [46] derived similar

bounds in terms of m on functions related to the Deque conjecture.

In [381, similar bounds were also derived for a different sequence extremal function. Let an (r, s)-formation be a concatenation of s permutations of r distinct letters. For example abcddcbaadbc is a (4, 3)-formation. Define Fr,s(n) to be the maximum length of any r-sparse sequence with n distinct letters which avoids all (r, s)-formations.

Klazar [30] proved that F,2(n) = O(n) and Fr,3(n) = O(n) for every r > 0. Nivasch

proved that F,,4(n) = 0(na(n)) for r > 2. Agarwal, Sharir, Shor [21 and Nivasch [381

showed that F,s(n) - n2 * )'1) for all r > 2 and odd s > 5 with t ₌ Y-2.

Let F,,,,k(m) be the maximum number of distinct letters in any sequence on m blocks avoiding every (r, s)-formation such that every letter occurs at least k times.

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Clearly F,,,,k(M) = 0 if m < k and F,,,,k(M) = oo if k < s and k K m. Every (r,

s)-formation contains an alternation of length s

+

1 for every r > 2, so AS+1,k(m)

Fr,,k(m) for every r > 2.

Nivasch proved for r > 2 that F,,4,2d+1(M) = O(mad(m)) for each fixed d > 2.

The recursive inequalities for the upper bounds on F,,4,2d+1 (in) in [38] also imply that

F,,4,6(M) = O(m log log

7n)

and F,4,2d+2(mn) = O(Mad(m)).

Similar bounds were also derived on an extremal function related to interval chains.

A k-chain on [1, m] is a sequence of k consecutive, disjoint, nonempty intervals of the

form [ao, a,][ai + 1, a2 ... [ak_1

+

1, ak] for integers 1 < ao K a1 < ... < ak < rM.

An s-tuple is a set of s distinct integers. An s-tuple stabs an interval chain if each element of the s-tuple is in a different interval of the chain.

Let (,,k(m) denote the minimum size of a collection of s-tuples such that every k-chain on [1, m] is stabbed by an s-tuple in the collection. Clearly ((,k(m) = 0 if

m <

k and (sk(m) is undefined if k

<

s and k K m.

Alon et al. [3] showed that 3,s(m) =

("

)

for s > 1, (3,4(m) O(mlogm),

(3,5(m) (m log logm), and (3,k(M) = (maLkj (M)) for k > 6.

Let rr,s,k(mi) denote the maximum size of a collection X of not necessarily distinct k-chains on [1, m] so that there do not exist r elements of X all stabbed by the same s-tuple. Clearly r,,,k(m) = 0 if m < k and 7r,,,k(m) = oo if k < s and k n.

In Section 2.1 we show that r,,,,,k(M) =F,sl+,k+l(m-i+ 1) for all r > 1 and 1 K s K

k K m. Since (.,k(i) rq2,,k(m) for all 1 K s K k K m, then AS+2,k+l(m+1) K (M)

for all 1 K s K _{k K m. This implies the bounds A,+}₁_,₈_{(in) <}

(" 7il)

_{for all s}_> _2,

A5,6(m) =

O(m

log log

m),

and A5,2d+2(M)= O(mad(in)) for d

>

3.

In Section 2.2 we construct alternation-avoiding sequences to prove lower bounds on As,k(m). We prove that A,+1,8(m) =

("

)

for all s > 2. Furthermore we show

that A5,6 (M) = Q(mloglogim) and A5,2d+2(m) = Q(jMad(M)) for d > 3. Thus the

bounds on A5,a(m) have a multiplicative gap of O(d) for all d.

2.1 Upper Bounds

We show that F,,s+1,k+l(m + 1) = rlr,s,(M) for all r 1 and 1 K s K k K M using

maps like those between matrices and sequences in [7] and [40].

Lemma 1. F,,sl+,k+l(M + 1) < Tlr,,k(m) for all r > 1 and 1 K s K k K m.

Proof. Let P be a sequence with Fr,s+l,k+1(m

+

1) distinct letters and m

+

1 blocks 1,.. ., _{m + 1 such that no subsequence is a concatenation of s + 1 permutations of}

r different letters and every letter in P occurs at least k + 1 times. Construct a

collection of k-chains on [1, m] by converting each letter in P to a k-chain: if the first

k + 1 occurrences of letter a are in blocks ao, ... , ak, then let a* be the k-chain with

ith interval [ai_ 1, a, - 1].

Suppose for contradiction that there exist r distinct letters q1,. .., q, in P such

that q*, ... , q* are stabbed by the same s-tuple 1 < Ji < ... <

j

₅K <m. Let

jo

= 0

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that b

c,i

[j1i- + 1, ji] for every 1 < n K r. Hence the letters qi,.. ., q, make an

(r, s

+

1)-formation in P, a contradiction. l

Corollary 2. AS+2,k+1(m + 1) K Q(m) for all 1 K s < k K m.

The bounds on (,,k(m) in

[3]

imply the next corollary.

Corollary 3. A,+i,8(m)

<

(

1 'jj

1 )

for

s

>

2, A5,5(m) O(mlogrm), A5,6(m) =

O(m log log m), and A5,k(m) = O(mak 21J(m)) for k > 7.

To prove that F ,,,+,k+1(M + 1) ms,k(m) for all r > 1 and 1 K s K k K m,

we convert collections of k-chains into sequences with a letter corresponding to each k-chain.

Lemma 4. Fr,s+1,k+1(m + 1) r,s,k(m) for all r > 1 and 1 K s < k K m.

Proof. Let X be a maximal collection of k-chains on [1, m] so that there do not exist r elements of X all stabbed by the same s-tuple. To change X into a sequence P

create a letter a for every k-chain a* in X, and put a in every block i such that either a* has an interval with least element i or a* has an interval with greatest element

i - 1.

Order the letters in blocks starting with the first block and moving to the last. Let Ai be the letters in block i which also occur in some block

j

< i and let Bi be

the letters which have first occurrence in block i.

All of the letters in Ai occur before all of the letters in Bi. If a and b are in Aj,

then a appears before b in block i if the last occurrence of a before block i is after the last occurrence of b before block i. The letters in Bi may appear in any order.

P is a sequence on m + 1 blocks in which each letter occurs k

+

1 times. Suppose

for contradiction that there exist r letters q, .. ., q,. which form an (r, s + 1)-formation

in P. List all (r, s + 1)-formations on the letters qi, .. ., q, in P lexicographically, so

that formation

f

appears before formation g if there exists some i > 1 such that the first i - 1 elements of f and g are the same, but the ith element of

f

appears before the ith element of g in P.

Let fo be the first (r, s + 1)-formation on the list and let 7ir (respectively pi) be the number of the block which contains the last (respectively first) element of the

ith permutation in fo for 1 i < s

+

1. Suppose for contradiction that for some 1 < i K s, 7i = Pi+1. Let a be the last letter of the ith permutation and let b be the

first letter of the (i + 1)st permutation.

Then a occurs before b in block ri and the b in 7i is not the first occurrence of b in P, so the a in wi is not the first occurrence of a in P. Otherwise a would appear after b in 7ri. Since the a and b in wi are not the first occurrences of a and b in P, then the last occurrence of a before 7i must be after the last occurrence of b before irr. Let

fi be the subsequence obtained by deleting the a in ir from fo and inserting the last

occurrence of a before 7i. Then fi is an (r, s

+

1)-formation and fi occurs before fo on the list. This contradicts the definition of fo, so for every 1 K i K s, 7ri < Pi+i.

For every 1

j

< r and 1 K i K s + 1, the letter qj appears in some block between pi and Tr inclusive. Since 7i < Pj+j for every 1 < i K s, the s-tuple ( 71,..., w) stabs

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each of the interval chains q*,.. , q,*, a contradiction. Hence P contains no (r, s +

1)-formation.

The idea for the next lemma is similar to a proof about doubled formation free matrices in [7].

Lemma 5. F,,s,s (m) < (r - 1)("- IV) for every r > 1 and I < s < m.

Proof. Let P be a sequence with m blocks such that no subsequence of P is a

con-catenation of s permutations of r distinct letters and every letter of P occurs at least

s times. An occurrence of letter a in P is called even if there are an odd number of

occurrences of a to the left of it. Otherwise the occurrence of a is called odd.

Suppose for contradiction that P has at least 1 + (r - 1) (7ni 1) distinct letters.

The number of distinct tuples (i, .... iLJ) for which a letter could have even

occurrences in blocks i,.. . , 2 j is equal to the number of positive integer solutions

to the equation (1 I x1) + ...

+

(1

+

xLj) + xI+LJ = + 1 if s is even and (1 +

x1) + ... + (1 + X + H) x1+[IJ = M if s is odd. Then by the pigeonhole principle there are at least r distinct letters q1, ... , q, with even occurrences in the same

[LJ

blocks. Then P contains a concatenation of s permutations of the letters q1,... q,

a contradiction. l

The last lemma is an alternate proof that A,+1,8(m) < ("r[l) since A8+,(rm) <

Fr,,,s(mfl) for all s, i > 1 and r > 2.

2.2 Lower bounds

In the last section we showed that A,+1,8(n)

(

for s > 2. The next lemma provides a matching lower bound.

Lemma 6. As+1,(n) >

("

71)

for all s > 2 and m > s + 1.

Proof. For every s > 1 and m > _{s + 1 we build a sequence X,(m) with} (j"-

₎

distinct letters. First consider the case of even s > 2. The sequence X,(rn) is the concatenation of i - 1 fans, so that each fan is a palindrome consisting of two blocks

of equal length.

First assign letters to each fan without ordering them. Create a letter for every -tuple of non-adjacent fans, and put each letter in every fan in its 1-tuple. Then order the letters in each fan starting with the first fan and moving to the last. Let Ai

be the letters in fan i which occur in some fan

j

< i and let B be the letters which

have first occurrence in fan i.

In the first block of fan i all of the letters in Ai occur before all of the letters in

Bi. If a and b are in Ai, then a occurs before b in the first block of fan i if the last

occurrence of a before fan i is after the last occurrence of b before fan i. If a and b are in Bi, then a occurs before b in the first block of fan i if the first fan which contains a without b is before the first fan which contains b without a.

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Consider for any distinct letters x and y the maximum alternation contained in the subsequence of X,(m) restricted to x and y. Start building the alternation with a fan that contains only x. Any other fans which contain x without y or y without x add at most 1 to the alternation length. Any fans which contain both x and y add 2 to the alternation length. If x and y occur together in i fans, then the length of their alternation is at most (s - i) + (I - i) + 2i = s.

Every pair of adjacent fans have no letters in common, so every pair of adjacent blocks in different fans can be joined as one block when the rn-1 fans are concatenated to form Xs(rn). Thus X,(m) has m blocks and ("2_) letters, and each letter occurs

s times.

For odd s > 3, construct X,(m) by adding a block r after Xs-1(m - 1) containing

all of the letters in X,-1(m - 1) such that a occurs before b in r if the last occurrence

of a in X,_(m - 1) is after the last occurrence of b in X,_1(m - 1). Then X,(m)

contains no alternation of length s + 1 since X,_1(m - 1) contains no alternation of

r q--+ 1

length s. Moreover X,(m) has m blocks and

("

)

letters, and each letter occurs 2

s times. l

The next lemma shows how to extend the lower bounds on As+,,,(m) to F,,,,,(m).

Lemma 7. F,s,k(m) (r - 1)F2,,,k(M) for all r > 1 and 1 < s < k < m.

Proof. Let P be a sequence with F2,,,k(m) distinct letters and m blocks such that no subsequence is a concatenation of s permutations of two distinct letters and every letter occurs at least k times. P' is the sequence obtained from P by creating r - 1 new letters a1, .. , a,-, for each letter a and replacing every occurrence of a with the

sequence a, ... a.

Suppose for contradiction that P' contains an (r, s)-formation on letters q₁,... , q,.

Then there exist indices i,

j,

k, 1 and distinct letters a, b such that qi = a3 and

qk= bl. P' contains a (2, s)-formation on the letters qi and qk, so P contains a (2, s)

formation on the letters a and b, a contradiction. Then P' is a sequence with (r

-1)F2,s,k(m) distinct letters and m blocks such that no subsequence is a concatenation

of s permutations of r distinct letters and every letter occurs at least k times. El

Corollary 8. Fr,,,, (m) = (r - 1)("

)

for all r > 1 and m > s > 1.

The proof of the next lemma is much like the proof that A5,2d+1(m) - Q(Mad(n))

for d > 2 in [381.

Lemma 9. A5,6(m) = Q(m log log m) and A5,2d+2(m) = Q(!mad(m)) for d > 3.

For all d, m > 1, we inductively construct sequences Gd(M) in which each letter appears 2d

+

2 times and no two distinct letters make an alternation of length 5. This proof uses a different definition of fan: fans will be the concatenation of two palindromes with no letters in common. Each palindrome consists of two blocks of equal length.

The sequences G1(m) are the concatenation of m + 1 fans. In each fan the second block of the first palindrome and the first block of the second palindrome make one

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block together since they are adjacent and have no letters in common. The first palindrome in the first fan and the second palindrome in the last fan are empty.

There is a letter for every pair of fans and the letter is in both of those fans. The letters with last appearance in fan i are in the first palindrome of fan i. They appear in fan i's first palindrome's first block in reverse order of the fans in which they first appear. The letters with first appearance in fan i are in the second palindrome of fan

i. They appear in fan i's second palindrome's first block in order of the fans in which

they last appear. By construction G1(m) contains no alternation of length 5.

For all d > 1 the sequence Gd(1) consists of 2d + 2 copies of the letter 1. The first and last copies of 1 are both special blocks, and there are empty regular blocks before the first 1 and after the last 1.

For d, m > 1 the blocks in Gd(m) containing only first and last occurrences of letters are called special blocks. Let Sd(m) be the number of special blocks in Gd(m).

Every letter has its first and last occurrence in a special block, and each special block in Gd(m) has m letters.

Blocks that are not special are called regular. No regular block in Gd(m) has special blocks on both sides, but every special block has regular blocks on both sides. The sequence Gd(m) for d, m > 2 is constructed inductively from Gd(m - 1) and Gd_1(Sd(m - 1)). Let

f

= Sd(m - 1) and g = Sd_1(f). Make g copies X1,..., X

of Gd(m - 1) and one copy Y of Gdl(f), so that no copies of Gd(m - 1) have any

letters in common with Y or each other.

Let Ai be the ith special block of Y. If the 1" element of Ai is the first occurrence of the letter a, then insert aa right after the Ith special block of Xi. If the 1 t" element

of Ai is the last occurrence of a, then insert aa right before the lPh special block of Xi. Replace A in Y by the modified Xi for every i. The resulting sequence is Gd(m).

Lemma 10. For all d and m, Gd(m) avoids ababa.

Proof. Given that the alternations in Gi(m) have length at most 4 for all m > 1, then

the rest of the proof is the same as the proof in [381 that Zd(m) avoids ababa. F-1 Let Ld(m) be the length of Gd(m). Observe that Ld(M) = (d + 1)mSd(m) since

each letter in Gd(m) occurs 2d + 2 times, twice in special blocks, and each special block has m letters.

Define Nd(m) as the number of distinct letters in Gd(m) and MAd(m) as the number

of blocks in Gd(m). Also let Xd(m) = Ad() _{Sd (M)} and Vlj(m) = Ld _Id(mf)". We bound _Xd(m)

and Vd,(m) as in [381.

Lemma 11. For all m, d > 1, Xd(mn) _{< 2d + 2 and V1(m) > M.}

Proof. By construction Si(m) = m

+

1 for m > 1, Sd(1) = 2 for d> 2, and Sd(7n) Sd(m -1)Sdl_(Sd(m-1)) for d, m > 2. Furthermore M1(m) = 3m-+3, MAd(1) = 2d+4

ford> 2, and Aid(m) =

A(m-)Sdl(Sd(m--1))+MAd(S(m-1))-Sdl_(Sd(m-1)) for d, m > 2.

Thus S2(m) = S2(m - 1)(S2(m - 1) + 1) 2(S2(m - 1))2 and S2(1) = 2. Since

S'(m) = 2 2-1 _{satisfies the recurrence S'(1)} = 2 and S'(m + 1) = 2(S'(m))2

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22m1 < S2(m) < 2 For d > 2, Sd(2) = 2Sd_1(2) and S1(2) = 3. So Sd(2) =

3 x ₂d-1

For d > 2, MAd(2) = (2d + 3)(3 x ₂d-2) + Mdl_(2) and M1(2) = 9. Hence MAd(2) =

(6d + 3)2d-1

Then X,1(m) = 3 for all m, Xd(1) d+2 for all d > 2, and Xd(2) = 2d+ 1 for all d > 2. For d, m > 2, Xd(m) = Xd(m - 1) + Xd-1(Sd(m-1))--1

Sd(M-1)

We prove by induction on d that Xd(m) 2d + 2 for all m, d > 1. Observe that the inequality holds for Xi(m), Xd(1), and Xd(2) for all m, d.

Fix d and suppose Xd1(m) < 2d for all m. Then Xd(m) Xdj(m - 1) +

s2-Hence Xd(m) < Xd(2) + (2d - 1) -2 Sd(n)) 1 = 2d + 1 + (2d - 1) -~=2 Sd(n)-1.

Since Sd(m) > 2Sd(m - 1) for all d, m > 2, then ' 2 Sd(n) )< 2SK(2)-l =

1 < 1 foral _ so Ld(m) -(d+1)MSd(rn) >

x2d-2 - 2d1 all d> , Xd(m) 2d+2. Hence Vd(m) -- m1 - (r)

The following analysis demonstrates the lower bounds on A5,2d+2 (m) for each

d >

2. If d = 2, let

mi =

M2(i) and

ni

= N2(i). Then mi

=

X2(i)S2(i)

<

6S2(i) <

6(221-1) < 22"+2 for i _-6> 1. Then i = Q(log log mi), so ni L₂(i) - V(i)N₆2(i) > _- i ₁₂2() =

Q(mi log log mi).

We use interpolation to extend the bound from mi to m. Let i and t satisfy mi <

m < mi+1 and t =

L[J.

Concatenate t copies of G2(i) with no letters in common for a total of at least [- Jni = Q(m log log m) letters. Hence A5,6(m) = Q(m log log mn).

We prove that S3(m) < A3(2m) following the method of [381. Since S3(n)

S3(m - 1)S2(S3(m - 1)) < S2(S3(n - 1))2

<

22s3(m-'+-2, then let F(in) = 22m+1_2

and G(m) = 22m. Then 2F(n) = 22m+_1 < 222, = G(2m) for every m > 0. Thus S3(m) < F(m-l)(S 3(1)) < 2F(M-n)(S 3(1)) 5 G(m-1)(2S3(1)) = A3(2mi).

Let mi = A13(i) and

ni =

N3(i). Therefore mi

=

X3(i)S3(i) K 8S3(i)

<

A3(2i

+

=_ 3()M i >3i iMV₃(i

2) for i > 1. So i = Q(a3(Mi)) and ni = N3(i) = ,> =

Q(mina3(mi)). Then A5,s(M) Q(ma3(m)) by interpolation.

For each d > 4 we prove that Sd(m) Ad(m + 2) for m > 1 by induction on d. Since S4(in) = S4(m - 1)S3(S4(m - 1)) < S3(S4(n - 1))2, then let F(m) = S3(im)2. Since 4F(m) < A3(4m) and A4(3) > 4S4(1), then S4(m) < F(rM- (S4(1)) <

4F(m-1) (S4(4)) < 4(fl-l)(4S4(1)) < A4(m + 2).

Fix d > 4 and suppose that Sd-_(m) < A-1 (m + 2). Define F(m) = Sd-(m)2. Since 4F(m) < Ad_1(4m) and Ad(3) > 4Sd(1), then Sd(m) < F(m-1) (Sd(1)) <

4F(m-1

)

(Sd(1))

<

A_" "(4Sd(1)) < 1

Ad(m

+ 2).

Fix d > 4. Let mi = Md(i) and ni = Nd(i). Then mi = Xd(i)Sd(i) < (2d +

2)Sd(i) < (2d + 2)Ad(i + 2) < Ad(i + 3). Then i > ad(Mi) - 3, so ni =di -Vd(i)Md(i) > iMA(i) = Q(lMiad(Mi)). By interpolation A,2d+2(m) = Q(1Mad(M)) for d > 4.

Corollary 12. For r > 2, F,4,6(m) = r.,3,(m) = Q(miloglognm) and Fr,4,2d+2(m)=

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2.3 Bounds on extremal functions of

0 -

1 matrices

using Davenport-Schinzel sequences

Let IF, (m, n) be the maximum length of a sequence with ?n blocks and n letters which avoids alternations of length s + 2. Nivasch showed that TV(m, n) < k(HFI'm) + n) for all k. We modify the matrix-sequence transformations in [401 to show bounds on exk(m, P) for alternating patterns P. Let P, be the 0 - 1 matrix with s rows

0, ... , s - 1 and 2 columns 0, 1 such that the number in each entry is the sum of its row and column mod 2.

Lemma 13. T, (m, n) = ex(n,n, P+1) for 'm, n, s > 1 and H(m) = exk(MPs+1) for s, k, m > 1.

Proof. The sequence to matrix transformation in [40] starts with a sequence

Q

with n letters on m blocks which avoids alternations of length s

+

2, and results with a

0 - 1 matrix A with n columns and m rows which avoids the matrix P,+. The letters

of Q are named 0,...,n -I by first occurrence and the entry in column i and row j of A is a one if and only if the letter i occurs in block

j

of

Q.

Suppose for contradiction that A contains P,+1. Then there is a submatrix of A with 2 columns co < ci and s + 1 rows ro, ... , r8 such that the entry in column ci and row rj is one if i +

j

is odd. The one entries in this submatrix correspond to an alternation c1co... of length s + 1 in

Q.

However, the first occurrence of co is before the first occurrence of c1 in

Q,

so

Q

contains an alternation of length s

+

2, a contradiction. This implies both T, (m, n) < ex(m, n, P,+1) for m, n, s > 1 and

Hs(M) exk(M, P+l) for s, k, m > 1.

Pettie used a matrix to sequence transformation to show that ex(m, n, P,+1) I(m, n)

+

n. 0 - 1 matrix A with m rows 0,..., m - I and n columns 0,. .,rn - 1

is converted to a sequence

Q

with m blocks 0,..., m - 1 and n letters 0,.. .,i - 1. Letter i occurs in block

j

of

Q

if and only if the entry of A in column i and row

j

is a one.

Let C be the letters in block

j

of

Q

which occur in no block before j and let Dj be the letters in block

j

of

Q

which occur in a block before j. All letters in Di occur before all letters in Cj in block

j,

and letters in Dj occur in block

j

in reverse order of their last appearance before block j. In [401 the letters in C were ordered arbitrarily, but here letter x in Ci occurs before letter y in Cj if and only if x < y.

Suppose that

Q

contains an alternation of length s + 2 on letters x and y such that

x < y. List all alternations of length s+2 on the letters x and y in

Q

lexicographically, so that alternation

f

appears before alternation g if there exists some i > 1 such that the first i - 1 elements of

f

and g are the same, but the i''h element of

f

appears before the ith element of g in

Q.

Let fo be the first alternation on the list and let 7i be the number of the block which contains the ith element of fo for 1 < i < s + 2. Suppose for the sake of contradiction that for some 2 < i K s + 1, i = wi+1. Let a be the ith element of fo

and let b be the (i + 1)'t element of fo.

Then a occurs before b in block ri and the b in 7ir is not the first occurrence of b in

Q,

so the a in 7i is not the first occurrence of a in Q. Otherwise a would appear after

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b in 1ri. Since the a and b in 7i are not the first occurrences of a and b in

Q,

then the last occurrence of a before 7i must be after the last occurrence of b before 7ri. Let

fi

be the subsequence obtained by deleting the letter a in 7i from fo and inserting the last occurrence of a before 7ri. Then fi is an alternation of length s +2 and fi occurs before fo on the list.

This contradicts the definition of fo, so for every 2 < i < s + 1, 7i < 7rj+1. We

now consider two cases. If the first element of fo is x, then the submatrix of A with 2 columns x, y and s + 1 rows 7r2, -- -,7r,+2 contains P+1 since x < y. If the first

element of fo is y, then suppose for contradiction that 7r, = r2. The occurrences of x and y in block 7r, are both first occurrences of x and y in Q. Otherwise, fo would not be the first alternation on the list. Since x and y are in Cr, then x appears before y in block 7r₁ because x < y, a contradiction. Then the submatrix of A with 2 columns x, y and s + 1 rows ,r+ 7r,... contains Ps+1.

Therefore ''(m, n) > ex(m, n, P+1) for m, n, s > 1 and I >-(m) exk(M, P+l)

for s, k, m > 1.

Corollary 14. ex,(m,Ps) =("j.) for all s > 2, ex6(m, P4) = (m loglog m), and

ex2d+2(m, P4) =

6

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Chapter 3 The formation width of sequences

If u is a sequence with r distinct letters, then let Ex(u, n) be the maximum length of

any r-sparse sequence with n. distinct letters that avoids u. If a and b are single letters,

then Ex(a, n) = 0, Ex(ab, n) = 1, Ex(aba, n) = n and Ex(abab, n) = 2n - 1. Nivasch [381 and Klazar [32] determined that Ex(ababa, n) ~ 2na(n). Agarwal, Sharir, and

Shor [21 proved the lower bound and Nivasch [381 proved the upper bound to show that if u is an alternation of length 2t + 4, then Ex(u, n) = n9nCdn)2O(a(l) t ') for

t > 1.

If u is a sequence with r distinct letters and c > r, then let Exc(u, n) be the

maximum length of any c-sparse sequence with n distinct letters which avoids u. Klazar [30] showed that Exc(u, n) = E(Exd(u, n)) for all fixed c, d > r.

Lemma 15. [30] For every sequence u with r distinct letters, Exd(u, n) Ex,(u, n)

(1 + Exc(u, d - 1))Exd(u, n) for all n > 1 and d > c > r.

An (r, s)-formation is a concatenation of s permutations of r distinct letters. For example abcddcbaadbc is a (4, 3)-formation.

Definition 16. F,s (n) is the maximum length of any r-sparse sequence with n

dis-tinct letters that avoids every (r, s)-formation.

Klazar [30] proved that Fr2 (n) = O(n) and F,3(n) = O(n) for every r. Nivasch

[38] proved that F,4(n) = O(na(n)) for r > 2. Agarwal, Sharir, and Shor

[2]

proved the lower bound and Nivasch [38] proved the upper bound to show that F,s()=

n2Aa()+!(a(n)-) for all r > 2 and odd s > 5 with t =

Nivasch [381 proved that Ex(u, n) < Fr,s-.r+i (n) for any sequence u with r distinct letters and length s by showing that every (r, s - r

+

1) formation contains u. Definition 17. The formation width of u, denoted by fw(u), is the minimum value

of s such that there exists an r for which every (r, s)-formation contains u. The

formation length of u, denoted by fl(u), is the minimum value of r such that every (r, fw(u))-formation contains u.

By Nivasch's proof,

fw(u)

< s - r + 1 for every sequence u with r distinct letters

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Lemma 18. If u contains v, then fw(v) fw(u).

Lemma 19. If u begins with the letter a, then fw(au) = fw(u) + 1.

Lemma 15 implies that fw(u) and fl(u) can be used to obtain upper bounds on

Ex(u, n).

Lemma 20. For any sequence u with r distinct letters and fixed integer c with c > r, Exc(u, it) = 0 (Ffl() (n))

-In this paper we use

fw(n)

primarily in order to prove tight upper bounds on

Ex (u, n) for several classes of sequences u such that a contains an alternation with

the same formation width as u. We also bound and evaluate fw for various other families of sequences in order to develop a classification of all sequences in terms of their formation widths.

If at is an alternation of length t for t > 2, then fw(at) < t - 1 since every

(r, t - 1) formation contains at for r > 2. Any (r, t - 2)-formation in which order of letters reverses in adjacent permutations avoids at, so fw(at) = t - 1. Pettie [411 used the fact that every (4,4)-formation contains abcacbc to prove the upper bound Ex(abcacbc, n) = O(na(n)). Since any (r, 3) formation with order

revers-ing in adjacent permutations would avoid abcacbc, then fw(abcacbc) = 4. Similarly

fw(abcadcbd) = 4.

Definition 21. An (r, s)-formation

f

is called binary if there exists a permutation p on r letters such that every permutation in

f

is either the same as p or the reverse of

p.

Most of the proofs in this paper depend on the fact that if u is a sequence with r distinct letters, then every binary (r, s)-formation contains u if and only if s > fw(u). We use the following notation to describe binary formations more concisely.

Definition 22. I is the increasing sequence 1 ... c on c letters and D, is the

decreas-ing sequence c ... 1 on c letters. Given a permutation 7r C Sc, the sequences I, and

D, are 7r(1) ... 7r(c) and 7(c) ... w(1) respectively.

We focus especially on two classes of binary formations in order to derive bounds on fw(u). The sequence up(l, t) is I, repeated t times, and alt(l, t) is a concatenation of t permutations, starting with I and alternating between I1 and D1. For example,

up(3, 3) = 123123123 and alt(3,3) = 123321123.

Definition 23. If u is a sequence with c distinct letters, then l(u) is the smallest k

such that up(c, k) contains a, and r(a) is the smallest k such that alt(c, k) contains

a.

Then fw(u) > l(a) and

fw(u)

> r(a). We evaluate both 1(u) and r(u) for every

binary formation U.

In Section 3.1 we prove that -y(r, s) = (r-1)2 _-₁₊₁_{is the minimum value for which} every (-y(r, s), s)-formation contains a binary (r, s)-formation. It follows that if u has

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r distinct letters, then fw(u) is the minimum s for which every binary (r, s)-formation

contains u.

In Section 3.3 we prove that fw(u) = t - 1 for every sequence u with two distinct letters and length t. We also determine every sequence u for which fw(u) < 3. In addition, we show that

fw(up(c,t))

= 2t - 1 for all c > 2 and t > 1. This implies that Ex(up(l, t), n) = n2 -- (n) t2s(Q(n) t 3) for all 1 > 2 and t > 3 and that

fw(u)

< 21(u) - 1 for every sequence u.

In Section 3.4.1 we compute l(u) and use the result to bound fw(u) up to a factor of 2 for every binary formation u. In particular we prove the following bounds on fw(u).

Theorem 24. Fix c > 2 and let u = I," D2

I 12 ... Cn, where L is I if n is odd and D if n is even, and ej > 0 for all i. Define A =

>

₁e2i-1 and B = E e2i. Let

M = max(A,B) and let m = min(A,B). Then (c- 1)m + M + [ n fw(u)

2(c - 1)m + 2M + 2[nJ - 1.

In Section 3.4.2 we compute r(u) for every binary formation u. Specifically we prove that if c > 2, then r(Ig1Dc2Ig3 ....Cen) = 2Z" e -nwhere L is I if n is odd

and D if n is even.

In Section 3.5 we use

fw(u)

to derive tight bounds on Ex(u, n) for other sequences

u besides up(l, t). Let u be any sequence of the form avav'a such that a is a letter, v is a nonempty sequence excluding a with no repeated letters and v' is obtained

from v by only moving the first letter of v to another place in v. We show that fw(u) = 4, implying that Ex(u, n) = E(na(n)). We also prove that Ex(abc(acb)t _,_n)₌

n 2 (Tt (n)(a(f)2 for all t > 2.

In Section 3.6 we compute fw for various classes of binary formations. In particular

we show for c > 2 and k > 1 that

fw(IcDcIc)

= c + 3,

fw(IcDc)

= c + 2k - 1,

fw(IcDcIcDc)

= 2c+3, fw(alt(c, 2k)) > k(c+2)-1, and

fw(alt(c,

2k+1)) > k(c+2)+1.

In Section 3.7 we discuss some unresolved questions.

3.1 An extension of the Erd6s-Szekeres theorem

The following upper bound is obtained by iterating the Erd6s-Szekeres theorem as in

[30].

Lemma 25. Every ((r - 1)2'-

+

1, s)-formation contains a binary (r, s)-formation. Proof. We prove by induction on s that every ((r - 1)2'-' + 1, s)-formation contains

a binary (r, s)-formation. Clearly this is true for s = 1. For the inductive hypothesis fix s and suppose for every r > 1 that each ((r - 1)281 + 1, s)-formation contains a

binary (r, s)-formation.

Consider any ((r - 1)2" +1, s+1)-formation F. Without loss of generality suppose that the first permutation of F is I(r_1)2-. By inductive hypothesis the first s

permutations of F contain a binary ((r-1)2_+1,

s)-formation

f.

By the Erd6s-Szekeres

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decreasing subsequence of length x. Therefore the last permutation of F contains an increasing or decreasing subsequence of length r on the letters of f. Thus F contains

a binary (r, s

+

1)-formation. l

Corollary 26. If u has r distinct letters, then every binary (r, s)-formation contains u if and only if s fw (u).

Proof. If for some s every binary (r, s)-formation contains t, then there exists a

function y(r, s) such that every (?(r, s), s)-formation contains i. Thus

fw(u)

< s.

If some binary (r, s - 1)-formation f avoids i, then for every z > r the binary (z, s - 1)-formations which contain f will avoid u. Hence

fw(u)

> s - 1. l Corollary 27. If i is a nonempty sequence and v is obtained from u by inserting a single occurrence of a letter which has no occurrence in i, then fw(u) =fw(v). Proof. If 't has r distinct letters, then every binary (2r + 1,fw(u))-formation F with

first permutation I2-+1 has a copy of u using only the even numbers 2,...,2r. Since there is at least one odd number between every pair of even numbers in F, then the copy of t in F can be extended to a copy of v using an odd number. El Corollary 28. If a has r distinct letters, then fl(u) < (r - 1)2ft (u+ 1.

Proof. Since every binary (r, fw(u))-formation contains u, then every ((r -1)2mu)-1 +

1,fw(u))-formation contains u. l

The next theorem shows that the upper bound in Lemma 25 is tight.

Theorem 29. For every r, s > 1 there exists a ((r - 1)2'1, s)-formation that avoids every binary (r, s)-formation.

Proof. We construct the desired formation Fa(r, s) one permutation at a time. Define

an a-block in Fa(r, s) to be a block of numbers in a permutation from positions

(k-1)(r-1)a+1 to k(r-1) for some k. For k < s-I define a k-swap on a permutation

of length (r - 1)2-1 as follows: For every even i, 1 < i < 2k, a k-swap reverses the

placement of the (i - 1)2'-k-'-blocks in each i2s-k-'-block. For example if (r, s) (3, 3), then a 1-swap on 1234567890ABCDEF produces CDEF90AB56781234.

Let permutation 1 of Fa(r, s) be the identity permutation on the letters 1, . . . , (r

-1)2-'. To form permutation k + 1 of Fa(r, s), perform a k-swap on permutation k.

The next lemma about Fa(r, s) will imply that Fa(r, s) avoids every binary (r, s)-formation.

Lemma 30. Consider any set B of distinct numbers occurring in each of the first k permutations of Fa(r, s) with the same or reverse order in adjacent permutations. Let i(k) = ej2k-j~1 where ej = 1 if the elements in B reverse order from

permutation j to permutation j + 1 and ej = 0 otherwise. Then in permutation k the elements of B are contained in different i(k)2sk-blocks, but the same (i(k) + 1)2

Bounds on extremal functions of forbidden patterns

Bounds on extremal functions of forbidden patterns

by

Jesse Geneson

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

z __J

(I

cc

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2015

@

Massachusetts Institute of Technology 2015. All rights reserved.

Signature redacted

A uthor ...

Department of Mathematics

May 10, 2015

Signature redacted

Certified by...

Peter Shor

Professor

Thesis Supervisor

Signature redacted

Accepted by ...

Michel Goemans

Chairman, Department Committee on Graduate Theses

Bounds on extremal functions of forbidden patterns

by

Jesse Geneson

Abstract

Acknowledgments

Contents

Chapter 1

Introduction

1.1

Matrix extremal functions

1.2

Sequence extremal functions

+

[1.

("F1)

1.3

Order of results

Chapter 2

The maximum number of distinct

letters in ababa-free sequences

4} for all k > 1.

+

7n)

+

m <

<

("

)

(" 7il)

O(m

m),

>

("

)

2.1

Upper Bounds

+

+

j

jo

c,i

+

[3]

<

(

1

'jj

1

)

for

>

j

₎