Vrije Universiteit Brussel
New values for the bipartite Ramsey number of the four-cycle versus stars Mattheus, Sam; Héger, Tamás; Hatala, Imre
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Discrete Mathematics
DOI:
10.1016/j.disc.2021.112320
Publication date:
2021
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Citation for published version (APA):
Mattheus, S., Héger, T., & Hatala, I. (2021). New values for the bipartite Ramsey number of the four-cycle versus stars. Discrete Mathematics, 344(5), [112320]. https://doi.org/10.1016/j.disc.2021.112320
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Discrete Mathematics
journal homepage:www.elsevier.com/locate/disc
New values for the bipartite Ramsey number of the four-cycle versus stars
Imre Hatala
a, Tamás Héger
b, Sam Mattheus
caGraduate, Mathematics BSc, ELTE Eötvös Loránd University, Budapest, Hungary
bELKH-ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Department of Computer Science, H-1117 Budapest, Pázmány P. sétány 1/C, Hungary
cDepartment of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium
a r t i c l e i n f o
Article history:
Received 26 September 2019
Received in revised form 12 January 2021 Accepted 18 January 2021
Available online 17 February 2021
Keywords:
Ramsey number Quadrilateral-free Projective plane Incidence graph Marginal set
a b s t r a c t
We provide new values of the bipartite Ramsey number RB(C4,K1,n) using induced subgraphs of the incidence graph of a projective plane. The approach, based on deleting subplanes of projective planes, has been used in related extremal problems and allows us to unify previous results and extend them. More importantly, using deep stability results on 2 modpsets and double blocking sets, we can show some of the limits of this technique when the projective plane is Desarguesian of large enough square order.
Finally, we also disprove two conjectures aboutRB(C4,K1,n).
©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Finite geometry, and in particular projective planes, has had many successful applications to (extremal) graph theory.
Among the questions that have been investigated are the Zarankiewicz problem [24], Turán numbers [16,17] and Ramsey theory [31,32]. The common thread in these problems is that they revolve around excluding the quadrilateral or four-cycle C4as a subgraph. One such problem is the following. ByKn,mwe denote the complete bipartite graph withnvertices in one of its vertex classes andmin the other one.
Definition 1.1. Thebipartite Ramsey numberRB(C4
,
K1,n) is the smallestbsuch that for every red-blue edge-coloring of Kb,bthere exists either a redC4or a blueK1,n.We will investigateRB(C4
,
K1,n) as a function innand denote this shortly byb(n). It is worthwhile to mention that this is a variation of the usual Ramsey numbersR(C4,
K1,n), which were recently the subject of study of Zhang, Cheng and Chen [31,32]. The bipartite counterpart had been investigated earlier by Carnielli, Gonçalves and Monte Carmelo in [10,21]where they obtained the following results.
Lemma 1.2([10]).For all n
≥
2, we have b(n)≤
n+
⌈√
n⌉
.
E-mail addresses: hatala.imre@gmail.com(I. Hatala),heger.tamas@ttk.elte.hu(T. Héger),sam.mattheus@vub.ac.be(S. Mattheus).
https://doi.org/10.1016/j.disc.2021.112320
0012-365X/©2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.
org/licenses/by-nc-nd/4.0/).
Using the incidence graphs of projective planes, and subgraphs thereof, they managed to obtain the following exact results. Note that the upper bound is attained for these values.
Theorem 1.3([10,21]).Let q be the order of a projective plane and1
≤
t≤
q+
1, then b(q2+
1)=
q2+
q+
2,
b(q2
+
1−
t)=
q2+
q+
1−
t.
In Section 2, we obtain more exact values for the functionb(n), using a more general method to obtain suitable subgraphs of incidence graphs of projective planes. This involves introducing a seemingly new (leastwise rather unknown) concept to graph theory which we callmarginal sets (Definition 2.7), and may be regarded as the counterpart of the well-known concept of dominating sets. Our method covers all of the above given values, and also provides several new instances; seeTheorem 2.12for details. Remark that all of these results attain the upper bound inLemma 1.2.
In Section3we investigate the method introduced in Section2to see whether we have exploited all of its capabilities.
The answer is partially affirmative: roughly speaking, our main result is that Desarguesian projective planes of large enough square order can provide only such examples ofb(n)
=
n+
⌈√
n⌉
that are also established byTheorem 2.12.
The values ofncovered byTheorem 2.12support the following conjecture of Carnielli and Monte Carmelo.
Conjecture 1.4(Conjecture 16, [10]).For all n, b(n)
=
n+ ⌈ √
n⌉
.Having formulated the above conjecture, the authors of [10] remark that ‘we do not expect to find any possible counter- example among the first values of the functionb(K2,2
;
K1,n)’, which isb(n) in their notation. A weaker conjecture was also formulated in [10].Conjecture 1.5(Conjecture 17, [10]).For all n, b(n)
<
b(n+
1).Clearly, ifConjecture 1.4holds, then so doesConjecture 1.5. However, in Section4we show that both conjectures fail, and we also provide new exact values ofb(n) that do not meet the upper bound ofLemma 1.2, the smallest such instance beingb(18)
=
22.In Section5we sum up the findings of the present paper, relate it to some other areas of (extremal) graph theory, and provide a table of the known exact values ofb(n) for smalln.
2. Constructions givingb(n)=n+⌈√ n⌉ To proveb(n)
=
n+
⌈√
n⌉
for somen, one has to ensure the existence of a bipartiteC4-free graphHwithn
+
⌈√
n⌉−
1 vertices in each part, not containingK1,nin its bipartite complement. Equivalently,Hneeds to have minimum degree at least⌈√
n⌉
. Then we may embedHinKn+
⌈
√n⌉
−1,n+⌈
√n⌉
−1and color all the edges ofHred, and all others blue. This shows thatb(n)>
n+
⌈√
n⌉
−
1 which, by the upper boundLemma 1.2, proves the desired result.2.1. Preliminaries
We will construct such graphs based on (finite) projective planes. We recall its definition and some notion here; for all others, such as the order of a projective plane, Baer subplanes, etc. we refer to [25]. All projective planes we discuss in the remainder will be finite, so we do not explicitly refer to them as such in what follows.
Definition 2.1. Aprojective planeΠ is a finite set of pointsPand linesLsuch that
•
every two lines meet in one point;•
every two points determine one line;•
there exist four points of which no three are collinear.If we omit the last condition in the definition of a projective plane, we obtain so-called degenerate projective planes.
An example of such a degenerate plane is thenear pencil, which contains three but no four points in general position. A near pencil may also be defined as follows.
Definition 2.2. Anear pencilof sizet
+
1 (t≥
2) consists oft+
1 pointsP0,
P1, . . . ,
Pt and thet+
1 lines{
P1,
P2, . . . ,
Pt}
,{
P0,
Pi}
(1≤
i≤
t). The pointP0will be called thepoint of concurrency.In the following, Πq and PG(2
,
q) will refer to an arbitrary, or to the Desarguesian projective plane of order q, respectively, whereasΠ will usually refer to a projective plane of arbitrary order.Definition 2.3. LetΠ be a projective plane with point setPand line setL. LetP′
⊂
PandL′⊂
L. Then (P′,
L′) is a closed systemofΠ if for any two points ofP′, the line ofΠjoining them is inL′and, dually, for any two lines ofL′, their intersection point is inP′.If we restrict the lines of a closed system to the point set of the system, then we obtain a (possibly degenerate) projective plane, hence we may call such a substructure a (degenerate) subplane. A closed system (P′
,
L′) of Π may be of the following four types:1.
|
P′| ≤
1 or|
L′| ≤
1.2. There exists an incident point-line pairP
∈
P′andℓ ∈
L′, and a non-empty setL′of lines throughPsuch thatℓ / ∈
L′, and a non-empty setP′of points onℓ
such thatP∈ /
P′, for whichP′= {
P} ∪
P′andL′= { ℓ } ∪
L′. 3. There exists a non-incident point-line pairP∈
P′ andℓ ∈
L′, and a setL′of two or more lines throughP, forwhichL′
= { ℓ } ∪
L′andP′= {
P} ∪ { ℓ ∩ ℓ
′:ℓ
′∈
L′}
.4. There exist four points in general position inP′, hence (P′
,
L′) is a non-degenerate subplane.In the first two types above, the number of points and lines may be different, while in the last two types they are equal. Let us note that closed systems of the third type are near pencils.
Definition 2.4. At-fold blocking setis a set of points that meets every line of the plane in at leastt points. Usually, 1-fold and 2-fold blocking sets are also called blocking and double blocking sets respectively. The dual concept for line sets are called (t-fold) covering sets.
Remark 2.5. Clearly, near pencils of sizet
+
1 (i.e. containingt+
1 points andt+
1 lines) are contained inΠqas long ast≤
q+
1. We recall that a non-degenerate subplaneΠ′with point setP′and line setL′of a projective planeΠqhas an ordersfor which eithers= √
q, ors2
+
s≤
qholds (this observation is attributed to Bruck). In the first case,Π′is a Baer subplane and|
P′| = |
L′| =
q+ √
q
+
1; in the latter case,s2+
s+
1= |
P′| = |
L′| ≤
q+
1. We say that a line belongs to a subplane if they have at least two points in common. Ifpis a prime power andt|
h, then PG(2,
ph) contains PG(2,
pt) as a subplane. Consequently, ifqis a square prime power, then PG(2,
q) contains Baer subplanes; moreover, it is well-known that PG(2,
q) can be partitioned intoq− √
q
+
1 pairwise disjoint Baer subplanes (so each point or line of PG(2,
q) belongs to exactly one Baer subplane of the partition). Finally, let us remark that the point set of a possibly degenerate subplaneΠ′ofΠ is a blocking set inΠ if and only if eitherΠ′is degenerate and contains all points of a line ofΠ, orΠ′is a Baer subplane ofΠ.Definition 2.6. Theincidence graph I(Πq) of a projective planeΠqwith point setPand line setLis the bipartite graph with partsPandL, where two vertices are adjacent if and only if they are incident in the projective plane. We will refer to the vertices in the two classes as points and lines as well.
Clearly,I(Πq) hasq2
+
q+
1 vertices in each part, it isC4-free, every vertex in it has degreeq+
1, and hence it does not containK1,q2+1 in its bipartite complement. This shows already thatb(q2+
1)>
q2+
q+
1, and by checking with Lemma 1.2, we see that this indeed givesb(q2
+
1)=
q2+
q+
2.
2.2. Constructions via marginal setsWe will construct induced subgraphs of I(Π) by deleting some vertices of it so that the minimum degree remains high, and we reach the upper bound ofLemma 1.2. As we want bipartite graphs with an equal amount of vertices in both parts, we need to delete as many points as lines. Hence we introduce the following concept, which will be central to our investigations.
Definition 2.7. LetGbe a graph with vertex setV. A proper subsetM of vertices is calledt-marginalif for all vertices u
∈
V\
M,uhas at mostt neighbors inM, andMis calledstrictly t-marginalif equality holds for some vertex not inM.IfGis bipartite with partsAandB, thenMis calledbalancedif
|
M∩
A| = |
M∩
B|
. In this case, thesizeofMis defined as|
M∩
A|
, and usually it will be denoted byv
.Note thatt-marginal sets are the counterpart oft-fold dominating sets, where each vertex not in the set is required to have at leasttneighbors in the set. We may call at-marginal setM perfectif each vertex not inMhas exactlytneighbors inM. Such sets are also called perfectt-fold dominating sets, which have their extremal graph theoretic applications as well; see Section5for further remarks. Let us point out a significant difference betweent-marginal andt-fold dominating sets: the family oft-marginal sets of a given graph is neither upward nor downward closed in general (with respect to containment), while any superset oft-fold dominating set is again at-fold dominating set.
Clearly, ifMis at-marginal set of ak-regular graphGwith vertex setV, thenV
\
Minduces a subgraph with minimum degree at leastk−
t. In the following, we will refer to the incidence graph of a planeΠ only by referring to the plane itself and, unless explicitly claimed otherwise, we will always consider non-empty and balancedt-marginal sets ofΠ. For sake of convenience, we will refer to them shortly by ‘t-marginal sets ofΠ’. By writingM=
P∪
L, we will always mean thatPis the set of points andLis the set of lines contained inM. Let us note thatt-marginal sets ofΠqare not interesting ift≥
q+
1.We will use the following observation without explicitly referring to it.
3
Remark 2.8. A setM
=
P∪
List-marginal inΠ if and only if every line ofΠnot inLmeetsPin at mostt points, and every point ofΠ not inPis covered by at mostt lines ofL.Lemma 2.9. Suppose that M
=
P∪
Lis a strictly t-marginal set ofΠq,1≤
t≤
q, and letv = |
P| = |
L|
. Then b(q2− v +
t+
1)≥
q2+
q+
2− v,
and
v ≤
2tq−
t2+
t.
Furthermore, the points and lines ofΠqnot in M induce a subgraph proving equality inLemma1.2if and only if
v ≥
2(t−
1)q−
t2+
3t,
and in this case we have b(q2
− v +
t+
1)=
q2+
q+
2− v
.Proof. The graphG spanned by theq2
+
q+
1− v
points and lines not inM isC4-free and its minimum degree is q+
1−
t. Consequently, there is noK1,q2+q+1−v−(q+1−t)+1=
K1,q2−v+t+1in the bipartite complement ofG, thus we obtain b(q2− v +
t+
1)≥
q2+
q+
2− v
.Lemma 1.2assertsb(n)≤
n+ ⌈ √
n
⌉
, whence q2+
q+
2− v ≤
q2− v +
t+
1+
⌈√q2
− v +
t+
1⌉
.
On the one hand, this means thatq
−
t<
√q2
− v +
t+
1 or, equivalently, (q−
t)2+
1≤
q2− v +
t+
1 must hold in all cases, which yields the upper bound stated. On the other hand, we obtain equality if and only if√q2
− v +
t+
1≤
q−
t+
1 or, equivalently,q2− v +
t+
1≤
q2+
t2+
1−
2qt+
2q−
2t, which leads to the asserted lower bound. □In the next two lemmas, we give constructions fort-marginal sets.
Lemma 2.10. Let M
=
P∪
Lbe a proper subset ofΠ. Then M is a1-marginal set ofΠ if and only if(P,
L)is a closed system ofΠ.Proof. IfMis 1-marginal, then the line connecting any two points ofPmust be inLand, dually, the intersection point of any two lines ofLmust be inP, henceMis a closed system. Conversely, ifMis a closed system, then for any pointP
∈ /
P, there can be at most one line ofLthroughP, otherwisePwould be the intersection point of two lines ofLwhence, asM is a closed system,P∈
Pwould follow, a contradiction. Dually, any line not inLmay contain at most one point ofP. □ Lemma 2.11. Let M1, . . . ,
Mtbe1-marginal sets of an arbitrary graph G. Then M1∪ · · · ∪
Mt is t-marginal. In particular, the union of t closed systems inΠ is t-marginal (not necessarily balanced or strict).Proof. Trivial. □
In Πq, a near pencil of size
v
exists for allv ∈ {
1, . . . ,
q+
2}
, which is a 1-marginal set inΠq byLemma 2.11and, clearly, it is strict. To satisfy the bounds ofLemma 2.9we need 2≤ v ≤
2q; hence near pencils of size 2≤ v ≤
q+
2 yieldb(q2+
2− v
)=
q2+
q+
2− v
, which gives backTheorem 1.3. However, we may use not only near pencils but the other types of closed systems ofΠq, in particular non-degenerate subplanes. In this way we also get graphs proving b(q2+
2− v
)=
q2+
q+
2− v
which are different from the previously found ones (except forv =
2), but as the only closed systems containing more thanq+
2 points are Baer subplanes (seeRemark 2.5), we obtain a new exact bipartite Ramsey number only in particular cases. Let us remark that the idea of removing a closed system from a projective plane is not new: it results in a so-called semi-plane, which is discussed already by Dembowski [13, Section 7.2]. However, we may take more than one closed systems to find new bipartite Ramsey numbers. In this way, we obtain the following theorem.Theorem 2.12. If there is a projective plane of order q (in particular, if q is a prime power), then b(q2
−
2q)=
q2−
q−
1,
andb(q2
−
2q+
1)=
q2−
q.
If there is a projective plane of order q that contains a Baer subplane (in particular, if q is a square prime power), then b(q2
−
q− √
q
+
1)=
q2− √
q
+
1,
and b(q2−
q− √
q
+
1−
s)=
q2− √
q
−
s for all q− √
q
≤
s≤
q.If there is a projective plane of order q that contains two distinct Baer subplanes that have s
≤
2√
q points in common (in particular, if q is a square prime power and s
∈
{0
,
1,
2,
3, √
q+
1, √
q
+
2} ), then b(q2−
2q−
2√
q
+
1+
s)=
q2−
q−
2√
q+
s.
Finally,
b(34)
=
40.
Proof. In all but the last case, we take one or the union of two closed systems of Πq as a 1- or 2-marginal setM (cf.Lemma 2.11); all instances will be clearly strict. Let us denote the number of the points ofMby
v
. CheckingLemma 2.9 we see thatMfits our needs if it is 1-marginal andv ≥
2, or 2-marginal andv ≥
2q+
2.Let
ℓ
1,ℓ
2 be two distinct lines, and let P1, / ∈ ℓ
1 and P2∈ / ℓ
2 be two distinct points inΠq. ThenPi andℓ
i define a near pencil of sizeq+
2 (i=
1,
2). Asℓ
1andℓ
2have one point in common, the union of these near pencils have sizev =
2q+
3−
c, where 0≤
c≤
2 is the number of incidences occurring amongP1∈ ℓ
2andP2∈ ℓ
1; note that we need c≤
1. ThenLemma 2.9assertsb(q2−
(2q+
3−
c)+
2+
1)=
b(q2−
2q+
c)=
q2+
q+
2−
(2q+
3−
c)=
q2−
q−
1+
c.This verifies the first two bipartite Ramsey numbers.
Suppose now that there exists a projective planeΠqcontaining a Baer subplaneΠ′(this is the case whenqis a square prime power, seeRemark 2.5). ThenΠ′itself is a strictly 1-marginal set of size
v =
q+ √
q
+
1, whenceLemma 2.9yields b(q2−
(q+ √
q
+
1)+
2)=
q2+
q+
2−
(q+ √
q
+
1), affirming the third assertion. Letℓ
be a line meetingΠ′in a single pointQ, and letPbe a point not inΠ′such that the linePQ belongs toΠ′(as there are√
q
+
1 lines throughQ belonging toΠ′, each containingq− √
qpoints not inΠ′, we can choose such a point). ThenPandsarbitrarily chosen points of
ℓ
different fromQ define a near pencilSof sizes+
1 such thatSandΠ′have no points or lines in common, so the strictly 2-marginal setΠ′∪
Shasv =
q+ √
q
+
1+
s+
1 points and lines. Assmay take any value betweenq− √
q andq(so thatv ≥
2q+
2), byLemma 2.9this provesb(q2−
(q+ √
q
+
2+
s)+
2+
1)=
q2+
q+
2−
(q+ √
q
+
2+
s), our fourth assertion.Suppose now thatΠqcontains two distinct Baer subplanesΠ1′,Π2′that haves
≤
2√
qpoints in common. Bose, Freeman and Glynn [7] proved that the common points and the common lines of two Baer subplanes always form a closed system with equally many points and lines, henceΠ1′
∪
Π2′has the same numberv =
2(q+ √
q
+
1)−
s≥
2q+
2 points and lines, and it is a strictly 2-marginal set.Lemma 2.9then yieldsb(q2−
(2q+
2√
q
+
2−
s)+
3)=
q2+
q+
2−
(2q+
2√
q
+
2−
s), our last assertion. Ifqis a square prime power, then PG(2,
q) is well-known to contain disjoint Baer subplanes; moreover, any four points in general position define a unique Baer subplane containing them, and if two Baer subplanes have three common points on a line, then all their√
q
+
1 points on that line are common. This means that the intersection of two Baer subplanes, which is a closed system as mentioned before, cannot contain four points in general position, hence it must be a degenerate subplane. If this degenerate subplane contains three collinear points, then it must contain√
q
+
1 collinear points; thus it may contain at most three points, or√
q
+
1 collinear points, or a near pencil of size√
q+
2. It is well-known that all these cases actually occur.Finally, if we take four disjoint Baer subplanes of PG(2
,
9) (recall that PG(2,
9) can be partitioned into 7 disjoint Baer subplanes), we obtain a 4-marginal set of size 52≥
2(4−
1)9−
42+
3·
4=
50, hence Lemma 2.9 gives b(81−
52+
4+
1)=
81+
9+
2−
52, as asserted. □Note that the union oft closed systems ofΠqhas at most
v =
t(q+ √
q
+
1) points, whereas we needv ≥
2(t−
1) q−
t2+
3t to meet the upper bound ofLemma 1.2, which clearly does not hold fort≥
3 ifqis large enough compared tot(precisely, ifq≥
15 and 3≤
t≤
q− √
q
−
1); but, as seen, we do get a bipartite Ramsey number not covered by the other constructions whenq=
9.3. Characterization
In this section our aim is to determine whether we have found all possible values ofnfor which we can get an exact value ofb(n) by finding a subgraph of a projective plane that yields equality inLemma 1.2. We start with some results for arbitrary planes, then proceed with Desarguesian ones by utilizing deep stability results on 2 modpsets and double blocking sets.
3.1. Combinatorial arguments for general planes
Recall that 1-marginal sets are completely characterized byLemma 2.10. Let us now give a general upper bound on the size oft-marginal sets that shows that takingt disjoint Baer subplanes is the best we can do in order to obtain a large enought-marginal set (to meet the lower bound ofLemma 2.9). We use the so-called incidence bound [28], which originally can be found in [23]. Recall that a setXalong with a familyBof its subsets, called blocks, is a 2-(
w,
k, λ
) design if|
X| = w
,∀
B∈
B:|
B| =
k, and any two distinct elements ofXare contained in preciselyλ
blocks. The replication number rof a design is the number of blocks containing a given point (this number does not depend on which point we take).Clearly,Πqis a 2-(q2
+
q+
1,
q+
1,
1) design with replication numberq+
1.Lemma 3.1([23,28]).Let(X
,
B)be a2-(w,
k, λ
)design with total number of blocks b and replication number r. For S⊆
X , T⊆
B, let i(S,
T)denote the number of incidences between the sets S and T , i.e. the cardinality of{
(x,
B)∈
S×
T|
x∈
B}
. Then we have⏐
⏐
⏐
⏐
i(S
,
T)−
r|
S||
T|
b⏐
⏐
⏐
⏐
≤ √
r− λ
√
|
S||
T|
(1
− |
S| w
) ( 1
− |
T|
b )
.
5
If equality occurs, then for each S′
∈ {
S,
X\
S}
and each T′∈ {
T,
B\
T}
, every point of S′is contained in a constant number of blocks of T′and every block of T′contains a constant number of points of S′.Theorem 3.2. Any t-marginal set M ofΠqhas size at most t(q
+ √
q
+
1). Furthermore, in case of equality, every line meets M in t or√
q
+
t points and, dually, every point is covered by t or√
q
+
t lines of M.Proof. LetM
=
P∪
Lbe at-marginal set ofΠq,|
P| = |
L| = v
. LetS=
PandT the set of all lines not inL. Then|
S| = v
,|
T| =
q2+
q+
1− v
andi(S,
T)≤
t(q2+
q+
1− v
) by the definition ofM. Apply the incidence bound (Lemma 3.1) withw =
b=
q2+
q+
1,r=
k=
q+
1,λ =
1 to get(q
+
1)v
(q2+
q+
1− v
)q2
+
q+
1−
t(q2+
q+
1− v
)≤
r|
S||
T|
b
−
i(S,
T)≤
√
r− λ
√
|
S||
T|
(1
− |
S| w
) ( 1
− |
T|
b )
=
√
q√
v
(q2+
q+
1− v
) (1
− v
q2
+
q+
1 ) (1
−
q2
+
q+
1− v
q2+
q+
1)
.
After clearing denominators, this leads to the inequality(q
+
1)v
(q2+
q+
1− v
)−
t(q2+
q+
1)(q2+
q+
1− v
)≤ √
q
v
(q2+
q+
1− v
),
from whichv
(q− √
q
+
1)≤
t(q2+
q+
1)=
t(q+ √
q
+
1)(q− √
q
+
1) follows. If equality holds, then every line not in LintersectsPintpoints and every line ofLmeetsPin some constantxpoints. Considert(q
+ √
q
+
1)(q+
1)= |
P|
(q+
1)=
t(q2+
q+
1− |
L|
)+
x|
L| =
t(q2+
q+
1−
t(q+ √
q
+
1))+
xt(q+ √
q+
1).
Simplifying byt(q+ √
q
+
1) and rearranging we getx=
(q+
1)−
(q− √
q
+
1−
t)= √
q
+
t. The other statement follows dually. □Since not-marginal set ofΠqcan be larger than the union oftdisjoint Baer subplanes, then, as pointed out previously, t-marginal sets can lead to graphs attaining equality inLemma 1.2only ift
≤
2 ort≥
q− √
q(underq
≥
15). Suppose that Gis aC4-free bipartite graph with minimum degreed≥
2. ThenGcan be considered as the incidence graph of a so-called partial linear space, and it can be easily embedded into a linear space. A well-known, long open and supposedly very difficult conjecture (see [5]) claims that every finite linear space can be embedded into a finite projective plane. Assume thatGcan be embedded intoΠq. Then, clearly, points and lines ofΠqnot belonging toGform a (q+
1−
d)-marginal set of Πq. Thus characterizingt-marginal sets ofΠqfor general values oft, or even fort≥
q− √
q, seems very difficult. Indeed, if we consider a subplaneΠsofΠq, its complement is a (q
−
s)-marginal set inΠqwiths≤ √
q; thus the aforementioned task includes describing all subplanes of projective planes. However, it is not known if every projective plane can be embedded into a larger (but still finite) one. Thus our aim in the following is to restrict ourselves tot
=
2 and study 2-marginal sets.Lemma 2.9andTheorem 3.2show that using a 2-marginal set ofΠqto obtain graphs that attain equality inLemma 1.2, we may get a bipartite Ramsey numberb(n) only ifq2
−
2q−
2√
q
+
1≤
n≤
q2−
2q+
1 for someqthat is the order of a projective plane. The next question is whether we find 2-marginal sets that prove equality for all values in this interval of length 2√
q
+
1, or just for the specific ones given by our constructions inTheorems 2.12. We believe that the second case is the truth, at least in PG(2,
q) ifqis large enough. Before proceeding with Desarguesian planes, we give some technical lemmas that hold for arbitrary planes. Ani-secant or (i+
)-secant line with respect to a point setPis a line that meetsP in exactlyior at leastipoints, respectively.Definition 3.3. LetPbe a fixed set of points inΠq.
•
For a pointP, letnPi (P) denote the number ofi-secant lines toPthroughP, and letnPi+(P) denote∑q+1 j=i nj(P).•
DefineLPi as the set of alli-secant lines toP, and letnPi:= |
Li|
. Also,LPi+andnPi+are defined analogously.IfPis clear from the context, we will drop the superscripts and writeni(P),ni+(P), etc.
Lemma 3.4. Fix an arbitrary point setPofΠq. If
|
P| ≥
2q+
3, thenL3+is a covering set.Proof. LetP
∈
P. Counting the points ofPon the lines throughP, we get 2q+
3≤ |
P| ≤
1+
(q+
1−
n3+(P))+
qn3+(P), which yieldsn3+(P)≥
2; that is, at least two lines ofL3+coverP. Let nowP∈ /
P. Considering the lines throughP, we get 2q+
3≤ |
P| ≤
2(q+
1−
n3+(P))+
qn3+(P), which yieldsn3+(P)≥
1; that is, at least one line ofL3+coversP. □We note that this lemma is sharp. Let
ℓ
1andℓ
2 be two distinct lines,P= ℓ
1∩ ℓ
2, and letQ∈ / ℓ
1∪ ℓ
2. ThenP=
ℓ
1∪ ℓ
2∪ {
Q}
has size 2q+
2, and sinceL3+consists ofℓ
1,ℓ
2and all lines throughQ exceptPQ, the points ofPQ\ {
P,
Q}
are not covered byL3+.Corollary 3.5. Let M
=
P∪
Lbe a2-marginal set ofΠqof sizev ≥
2q+
3. ThenPis a blocking set andLis a covering set.Proof. SinceMis 2-marginal,P3+
⊆
PandL3+⊆
Lhold, hence the assertion follows immediately fromLemma 3.4and its dual. □Lemma 3.6. Let M
=
P∪
Lbe a2-marginal set ofΠqwith|
P| = |
L| = v ≥
2q+
3. Define c by|
P| =
2q+
2√
q
+
2−
c.Then
n1
≤
c(2q√
q−
2q−
2√
q+
c) 2√
q
+
1,
andn1
+
n3+≤
c(2q√
q−
2q−
2√
q+
c) 2√
q
+
1+
2q+
2√
q
+
2−
c,
where niand ni+are the quantities defined inDefinition3.3with respect toP.
Proof. ByTheorem 3.2andCorollary 3.5,c
≥
0 andn0=
0. AsM is 2-marginal,n3+≤ v
clearly holds. Let us write a slightly rearranged form of the so-called standard equations:q+1
∑
i=3
ni
=
q2+
q+
1−
n1−
n2,
q+1
∑
i=3
ini
= v
(q+
1)−
n1−
2n2,
q+1
∑
i=3
i(i
−
1)ni= v
(v −
1)−
2n2.
Then we have0
≤
q+1
∑
i=3
(i
−
(√
q
+
2))2ni=
q+1
∑
i=3
i(i
−
1)ni−
(2√
q+
3)q+1
∑
i=3
ini
+
(√
q+
2)2q+1
∑
i=3
ni
= v
(v −
1)−
2n2−
(2√
q
+
3)(v
(q+
1)−
n1−
2n2)+
(√
q
+
2)2(q2+
q+
1−
n1−
n2),
equivalently,(q
+
2√
q
+
1)n1+
qn2≤ v
(v −
1)−
(2√
q
+
3)v
(q+
1)+
(√
q
+
2)2(q2+
q+
1).
It follows thatq(n1
+
n2)+
(2√
q
+
1)n1≤ v
(v −
1)−
(2√
q
+
3)v
(q+
1)+
(√
q
+
2)2(q2+
q+
1).
Let us usev ≥
n3+=
q2+
q+
1−
n1−
n2to findq(q2
+
q+
1− v
)+
(2√
q
+
1)n1≤ v
(v −
1)−
(2√
q
+
3)v
(q+
1)+
(√
q
+
2)2(q2+
q+
1),
that is,n1
≤ v
(v −
1)−
(2√
q
+
3)v
(q+
1)+ v
q+
4(√
q
+
1)(q2+
q+
1) 2√
q
+
1.
If we write
v =
2q+
2√
q
+
2−
c, we obtain n1≤
c(2q√
q−
2q−
2√
q+
c) 2√
q
+
1.
Furthermore, asn3+
≤ v =
2q+
2√
q
+
2−
c, n1+
n3+≤
c(2q√
q−
2q−
2√
q+
c) 2√
q
+
1+
2q+
2√
q
+
2−
c.
□Note that the upper bound found on n1
+
n3+is increasing in cunder 0≤
c≤
2√
qifq
≥
7. Let us remark that Theorem 3.2can also be proved with the help of the standard equations, yet the proof is much more compact if we use the incidence bound.7
3.2. Arguments forPG(2
,
q)Throughout this part,M
=
P∪
Ldenotes a 2-marginal set of PG(2,
q) with|
P| = |
L| = v ≥
2q+
3. ByCorollary 3.5 andLemma 3.6, we see that ifv
is not too small, then almost all lines intersectPin exactly two points. Our strategy is to slightly modifyPso that it becomes a weighted double blocking set (see definition below), and then use characterization results on these types of sets.Definition 3.7. Amultiset orweighted setS will be considered as a set of points equipped with non-negative integer weights. Thesizeortotal weightofS, in notation
|
S|
, is the sum of the weights of the points inS. The weighted intersection of a lineℓ
withSis the sum of the weights of the points inℓ ∩
S. If we say that a line intersects a multisetSinkpoints, we mean that its weighted intersection withSisk. Aweighted k-fold blocking setis a weighted set which intersects every line in at leastkpoints. Weighted 2-fold blocking sets are also called weighted double blocking sets.First we will use a theorem of [30] about multisets of PG(2
,
q) which intersect almost all lines ink modppoints, wherepis the characteristic of the underlying field GF(q), andkis an arbitrary integer.Theorem 3.8([30], Theorem 3.6).LetSbe a multiset inPG(2
,
q),17<
q, q=
ph, where p is prime. Let0≤
k<
p be an arbitrary integer. Assume that the number of lines intersectingSin non-(k modp) points isδ
, whereδ <
(⌊ √
q
⌋ +
1)(q+
1−
⌊ √
q
⌋
). Assume furthermore that the following property holds: if through a point P there are more than q/
2lines intersecting Sin non-k modp points, then there exists a value r(P)such that the weighted intersection of more than q2+δ1+
5of these lines withSis r(P) modp. Then there exists a multisetS′with the property that it intersects every line in k modp points and the number of different points in(S∪
S′)\
(S∩
S′)is exactly⌈ δ q+1
⌉ .
The statement of the above theorem may be rephrased in the following way: there exists a setXof
⌈ δ q+1
⌉
points such that we can obtain the multisetS′fromSby giving appropriate weights of the points inX, and leaving the weights in S
\
Xuntouched. (Note thatX⊂
Sis not necessarily true.)We setk
=
2 to applyTheorem 3.8forP, considered as a multiset where every point has weight one. Letδ
denote the total number of non-(2 modp) secant lines toP; thenδ ≤
n1+
n3+(recalln0=
0), the quantity bounded from above byLemma 3.6. To meet the requirementδ <
(⌊ √
q
⌋ +
1)(q+
1− ⌊ √
q
⌋
) (1)ofTheorem 3.8, we distinguish two cases. Ifqis a square, we need
δ <
q√
q
+
1. Ifqis not a square, then substituting⌊ √
q⌋ = √
q
− ε
into(1)(0≤ ε <
1), it is easy to see that the right-hand side is strictly decreasing inε
ifq≥
4. Thus it is enough to satisfy the inequality forε =
1, that is, to haveδ ≤
q√
q
−
q+
2√
q.Let
|
P| =
2q+
2√
q
+
2−
cas inLemma 3.6. Ifqis a square, we assumec≤ √
q
−
1 (that is,|
P| ≥
2q+ √
q+
3, which is the smallest possible size of a 2-marginal set not covered by the constructions inTheorem 2.12), in which case Lemma 3.6yieldsδ ≤
n1+
n3+≤
(√
q−
1)(2q√
q
−
2q− √
q−
1) 2√
q
+
1+
2q+ √
q
+
3≤
q√
q,
provided thatq≥
49. Ifqis not a square, we assumec≤ √
q
−
2 (that is,v ≥
2q+ √
q
+
4; this is needed to obtain a suitable upper bound onδ
). Then, underq≥
32,Lemma 3.6similarly yieldsδ <
q√
q
−
q+
2√
q.
Thus the respective assumptions oncenable us to applyTheorem 3.8forP as soon as it has the other property. The quantity involved therein,q2+δ1
+
5, can be bounded above by 2√
q
+
5 in both cases. Suppose now that for a pointP, the number of non-(2 modp) secants throughPtoPis more thanq/
2, then clearlyn2(P)≤
(q+
1)/
2. IfP∈ /
P, there are at most two (3+
)-secants toPonP. Recallingn0(P)=
0 byCorollary 3.5, thenn1(P)≥
q+
1−
2−
(q+
1)/
2=
(q−
3)/
2>
2
√
q
+
5 holds ifq≥
61. Suppose nowP∈
P. Then, for any integert≥
2, we have|
P| ≥
1+
t−1
∑
i=1
ni(P)(i
−
1)+
(t−
1)nt+(P).
Asnt+(P)=
q+
1−
∑t−1i=1ni(P), this yields
|
P| ≥
1+
(t−
1)(q+
1)−
t−1
∑
i=1
(t
−
i)ni(P)=
1+
(t−
1)(q+
1)−
(t−
2)n2(P)−
t−1
∑
i=1 i̸=2
(t
−
i)ni(P).
Suppose now and substituteni
≤
2√
q
+
5 for all 1≤
i≤
t−
1,i̸=
2 to get|
P| ≥
1+
(t−
1)(q+
1)−
(t−
2)(q+
1)2
−
((t 2 )
−
(t−
2) )(2
