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CHAPTER 2
Ramsey theory
2.1. Introduction
In this chapter, we will survey graph (and hypergraph) problems of Paul Erd˝os (often with his collaborators) arising out of his work in Ramsey theory. The guiding philosophy in this subject deals with the inevitable occurrence of specific structures in some part of a large arbitrary structure which has been partitioned into finitely many parts. Well-known examples of this are the Pigeonhole Principle, van der Waerden’s theorem on arithmetic progressions and Ramsey’s theorem itself. We will say more about these in subsequent sections.
2.2. Origins
Paul’s first results in this area occurred in his joint paper 1 , written with George Szekeres and published in 1935. Simply titled, “A combinatorial problem in geom- etry”, it laid the groundwork for an amazing variety of subsequent work during the next 60 years. This question arose out of a question posed by Esther Klein, a talented young mathematician in Budapest, who asked:
Is it true that for all n, there is a least integer g(n) so that any set of g(n) points in the plane in general position must always contain the vertices of a convex n-gon?
She had previously observed that g(4) = 5. The reader is encouraged to read Szekeres’ touching accounts 2 3^ of how this joint work arose, and the effects it had on his life and career (in particular, he married Esther Klein the following year, in 1936, and they remain still happily married, living and working in Australia now. This is the reason Paul often referred this affirmative solution to Esther Klein’s question as the “Happy End” theorem.)
In proving that g(n) exists, Szekeres actually rediscovered Ramsey’s theorem, which had only appeared (unknown to him then) some five years earlier. Erd˝os and
(^1) P. Erd˝os and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470. (^2) Paul Erd˝os, The Art of Counting, ed. Joel Spencer, The MIT Press, Cambridge, Mas- sachusetts, 1973. (^3) P. Erd˝os, Some of my favorite problems and results, The Mathematics of Paul Erd˝os (R. L. Graham and J. Neˇsetˇril, eds.), 47–67, Springer-Verlag, Berlin, 1996.
5
6 2. RAMSEY THEORY
Szekeres established the following bounds on g(n):
(2.1) 2 n−^2 + 1 ≤ g(n) ≤
2 n − 4 n − 2
They further conjectured that the lower bound is actually the correct answer.
The proof for (2.1) is based on several interrelated fundamental facts which illustrate the spirit of Ramsey theory:
(i) For any sequence of n^2 + 1 distinct numbers, say, x 1 , x 2 ,... , xn (^2) +1, there is always either an increasing subsequence (i.e., xi 1 ≤ xi 2 ≤... ≤ xin+1 with i 1 < i 2 <... < in+1) of n + 1 numbers, or a decreasing subsequence (i.e., xj 1 ≥ xj 2 ≥
... ≥ xjn+1 with j 1 < j 2 <... < jn+1) of length n + 1.
(ii) For given positive integers m and n, any set of
(n+m− 2 n− 1
- 1 points in general position in the plane must contain either n points x 1 ,... , xn with consecutive line segments xixi+1 of increasing slopes, or m points with consecutive line segments of decreasing slopes.
Both (i) and (ii) have short elegant proofs which are perhaps the Book Proofs. In Erd˝os’ language, those proofs belong to the Book (containing the best possible proofs of each theorem in mathematics), which we mortals can only occasionally get a glimpse of.
Proof of (i): We associate to each number xj , a pair of integers (aj , bj ) where aj denotes the length of the longest increasing subsequence ending at xj , and bj denotes the length of the longest decreasing subsequence ending at xj. It is easy to see that (ai, bi) 6 = (aj , bj ) for i 6 = j. Since there are n^2 + 1 numbers xj , not all the (aj , bj ) can satisfy aj , bj ≤ n. Thus, there is a monotone subsequence of length at least n + 1.
Proof of (ii): Let f (n, m) denote the maximum number of points such that there is no n-cup (i.e., n points with consecutive line segments having increasing slopes), and there is no m-cap (i.e., m points with consecutive line segments having decreasing slopes). It suffices to show
f (n, m) ≤ f (n, m − 1) + f (n − 1 , m).
Suppose S is a set of f (n, m) points containing no n-cup and no m-cap. We consider the set T of points x which are the right endpoints of some (n − 1)-cup. Clearly, x cannot be the left endpoint of an (m − 1)-cap. Therefore, we have
|T | ≤ f (n, m − 1).
Also, |S \ T | ≤ f (n − 1 , m).
This proves (ii).
Now the upper bound for g(n) follows immediately from (ii) since an n-cup or n-cap forms a convex n-gon.
The lower bound for g(n) in (2.1) is established by appropriately combining sets of sizes f (bn/ 2 c − 2 i, bn/ 2 c + 2i) for integers i in the interval (−bn/ 2 c, dn/ 2 e).
8 2. RAMSEY THEORY
2.3. Classical Ramsey Theory
Here we state the simple version of Ramsey’s theorem for coloring graphs in two colors. The original statement is much more general. The versions for hypergraphs, infinite graphs and/or with more colors will be discussed in later sections.
For two graphs G and H, let r(G, H) denote the smallest integer m satisfying the property that if the edges of the complete graph Km are colored in red or blue, then there is either a subgraph isomorphic to G with all red edges or a subgraph isomorphic to H with all blue edges.
The classical Ramsey numbers are those for the complete graphs and are de- noted by r(s, t) = r(Ks, Kt). In the special case that n 1 = n 2 = n, we simply write r(n) for r(n, n), and we call this the Ramsey number for Kn.
2.3.1. On Ramsey numbers for Kn. The problem of accurately estimating r(n) is a notoriously difficult problem in combinatorics. The only known values 11 are r(3) = 6 and r(4) = 18. For r(5), the best bounds 12 ,^13. are 43 ≤ r(5) ≤ 49. For the general r(n), the earliest bounds were:
(2.4)
e
n 2 n/^2 < r(n) ≤
2 n − 2 n − 1
The upper bound follows from the fact that the Ramsey number r(k, l) satisfies:
(2.5) r(k, l) ≤ r(k − 1 , l) + r(k, l − 1)
with strict inequality if both r(k − 1 , l) and r(k, l − 1) are even. To see this, if n = r(k − 1 , l) + r(k, l − 1), for any vertex v, there are either at least r(k − 1 , l) red edges or at least r(k, l − 1) blue edges leaving v. Therefore there is either a red copy of Kk or a blue copy of Kl. The strict inequality condition is a consequence of the fact that a graph on an odd number of vertices can not have all odd degrees.
The lower bound is established by a counting argument given by Erd˝os in 14 , which can be described as follows:
There are 2(
m 2 ) ways to color the edges of Km in 2 colors. The number of
colorings that contain a monochromatic Kn is at most ( m n
m 2 )−( n 2 )+1.
Therefore, there exists a coloring containing no monochromatic Kk if
2 (
m 2 )^ >
m n
m 2 )−( n 2 )+1.
(^11) R. E. Greenwood and A. M. Gleason, Combinatorial relations and chromatic graphs, Canad. J. Math. 7 (1955), 1-7. (^12) G. Exoo, A lower bound for R(5, 5), J. Graph Theory 13 (1989), 97-98. (^13) B. D. McKay and S. P. Radziszowski, Subgraph counting identities and Ramsey numbers, J. Comb. Theory (B), 69 (1997), 193-209. (^14) P. Erd˝os, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.
2.3. CLASSICAL RAMSEY THEORY 9
This is true when
m ≥
e
n 2 n/^2.
So, the lower bound in (2.4) is proved.
Very little progress has occurred in the intervening fifty years in improving these bounds. The best current bounds are
(2.6)
e
n 2 n/^2 < r(n) < n−^1 /2+c/
√log n^ ( 2 n − 2 n − 1
The upper bound is due to Thomason 15 and the lower bound is due to Spencer 16 by using the Lov´asz local lemma, which we will describe here.
The Lov´asz local lemma
Let A 1 ,... , Aq be events in an arbitrary probability space. Suppose that each event Ai is mutually independent of a set of all but at most d of the other events Aj , and that P r(Ai) ≤ p for all 1 ≤ i ≤ q. If
ep(d + 1) ≤ 1 ,
then P r(
∧q i=1 A¯i)^ >^ 0. For each set S of n vertices in a graph with m vertices, let AS denote the event
that the complete graph on S is monochromatic. Therefore, P r(AS ) = 2^1 −(
n 2 ) =
p. Since each event AS is mutually independent of all the events AT satisfying |S ∩ T | ≤ 1, we have d =
(n 2
)( (^) m n− 2
. Using the Lov´asz local lemma, if
e(
n 2
m n − 2
+ 1)2^1 −(
n 2 )^ < 1 ,
we have r(n, n) > m. A straightforward simplification gives
r(n) >
e
n 2 n/^2.
In particular, we see that r(n)^1 /n^ lies between
2 and 4.
Conjecture $100 (1947) The limit (2.7) (^) nlim→∞ r(n)^1 /n exists.
Problem $250 (1947) Determine the value of (2.8) c := lim n→∞ r(n)^1 /n
if it exists. (^15) A. Thomason, An upper bound for some Ramsey numbers, J. Graph Theory 12 (1988), 509–517. (^16) J. Spencer, Ramsey’s theorem—a new lower bound, J. Comb. Theory Ser. A 18 (1975), 108–115.
2.3. CLASSICAL RAMSEY THEORY 11
implies that G contains no clique or independent set of size
m q − 1
. By choos-
ing m = q^3 , we obtain a graph on n =
m q^2 − 1
vertices containing no clique or
independent set of size ec(log^ n^ log log^ n) 1 / 2 .
In the past ten years, there has been a great deal of development in explicit constructions of so-called expander graphs. (which are graphs with certain isoperi- metric properties). In particular, Lubotzky, Phillips and Sarnak 22 and Margulis^23 24 25 (^) have successfully obtained explicit constructions for expander graphs. How-
ever, we are still quite far away from constructing Ramsey graphs on n vertices which contain no clique of size c log n and no independent set of size c log n.
2.3.3. Off-diagonal Ramsey numbers. For off-diagonal Ramsey numbers, the additional known values are r(3, 4) = 9, r(3, 5) = 14, r(3, 6) = 18, r(3, 7) = 23, r(3, 8) = 28, r(3, 9) = 36 and r(4, 5) = 25 while 35 ≤ r(4, 6) ≤ 41 (see the dynamic survey of Radzisowski on small Ramsey numbers in the Electronic Journal of Combinatorics, at www.combinatorics.org for more bounds and references).
For k = 3, Kim 26 recently proved a new lower bound which matches the previous upper bound for r(3, n) (up to a constant factor), so it is now known that
(2.11)
cn^2 log n
< r(3, n) < (1 + o(1))
n^2 log n
Ajtai, Koml´os and Szemer´edi 27 earlier gave the upper bound of c′^ n
2 log n and Shearer 28 29 (^) replaced c′ (^) by 1 + o(1). It would be of interest to have an asymptotic formula
for r(3, n).
The best known constructive lower bound for r(3, n) is due to Alon 30
r(3, n) ≥ cn^3 /^2.
(^22) A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988), 261–
(^23) G. A. Margulis, Arithmetic groups and graphs without short cycles, 6th Internat. Symp. on Information Theory, Tashkent (1984) Abstracts 1 , 123-125 (in Russian). (^24) G. A. Margulis, Some new constructions of low-density parity check codes, 3rd Internat. Seminar on Information Theory, convolution codes and multi-user communication, Sochi (1987), 275-279 (in Russian). (^25) G. A. Margulis, Explicit group theoretic constructions of combinatorial schemes and their applications for the construction of expanders and concentrators, Problemy Peredaci Informacii (1988) (in Russian). (^26) J. H. Kim, The Ramsey number R(3, t) has order of magnitude t (^2) / log t, Random Structures and Algorithms 7 (1995), 173–207. (^27) M. Ajtai, J. Koml´os and E. Szemer´edi, A note on Ramsey numbers, J. Comb. Theory Ser. A 29 (1980), 354–360. (^28) J. Shearer, A note on the independence number of triangle-free graphs, Discrete Math. 46 (1983), 83-87. (^29) J. Shearer, A note on the independence number of triangle-free graphs II, J. Comb. Theory (B) 53 (1991), 300-307. (^30) N. Alon, Explicit Ramsey graphs and orthonormal labellings, Elec. J. Comb. 1 (1994), R12 (8pp).
12 2. RAMSEY THEORY
improving previous bounds of Erd˝os^31 and others 32.
For r(4, n), the best lower bound known is c(n log n)^5 /^2 due to Spencer, 33 again by using the Lov´asz local lemma. The best upper bound known is c′n^3 / log^2 n, proved by Ajtai, Koml´os and Szemer´edi^27. So there is a nontrivial gap still remain- ing, as repeatedly pointed out in many problems papers 34 of Erd˝os.
Problem 19 ($250) Prove or disprove that
(2.12) r(4, n) >
n^3 logc^ n for some c, provided n is sufficiently large. For general k, the best asymptotic bounds for r(k, n), for n large, are as follows:
(2.13) c
n log n
)(k+1)/ 2 < r(k, n) < (1 + o(1))
nk−^1 logk−^2 n
The upper bound is a recent result of Li and Rousseau 35 who extend Shearer’s method to improve the constant factor for the bounds in 27. The lower bound is given in 33.
Conjecture (1947) For fixed k,
(2.14) r(k, n) > nk−^1 logc^ k for a suitable constant c > 0 and n large.
Very few results are known about the gaps between ‘consecutive’ Ramsey num- bers. Here are several problems appearing in the 1981 problem paper^36.
Problem (Burr and Erd˝os^36 ) Prove that (2.15) r(n + 1, n) > (1 + c)r(n, n) for some fixed c > 0.
(^31) P. Erd˝os, On the construction of certain graphs, J. Comb. Theory 17 (1966), 149- (^32) F. R. K. Chung, R. Cleve and P. Dagum, A note on constructive lower bounds for the Ramsey numbers R(3, t) (^33) J. Spencer, Asymptotic lower bounds for Ramsey functions, Discrete Math. 20 (1977/78), 69–76. (^34) P. Erd˝os, Problems and results on graphs and hypergraphs: similarities and differences, Mathematics of Ramsey theory, Algorithms Combin., 5, (J. Neˇsetˇril and V. R¨odl, eds.), 12–28, Springer, Berlin, 1990. (^35) Y. Li and C. C. Rousseau, Bounds for independence numbers and classical Ramsey num- bers, preprint. (^36) P. Erd˝os, Some new problems and results in graph theory and other branches of combina- torial mathematics, Combinatorics and graph theory (Calcutta, 1980), Lecture Notes in Math., 885 , 9–17, Springer, Berlin-New York, 1981.
14 2. RAMSEY THEORY
This original problem has been settled in the affirmative by Chv´atal, R¨odl, Szemer´edi and Trotter 38. Their proof is a beautiful illustration of the power of the regularity lemma of Szemer´edi.
Roughly speaking, the regularity lemma says that for any graph G, we can partition G into a relatively small number of parts such that the bipartite graph between most pairs of parts behaves like a random graph. To be specific, a bipartite graph with vertex set A ∪ B is said to be -regular if for any X ⊂ A and Y ⊂ B with |X| ≥ |A|, |Y | ≥ |B|, the edge density in the induced subgraph X ∪ Y is essentially the same as the edge density in A ∪ B (differs by at most an additive term of ).
The bounded number of parts depends only on and is independent of the size of G. The main part of the proof 38 is accomplished by repeatedly using the -regular property to find a desired monochromatic subgraph. (For an excellent survey article on the regularity lemma and its many applications, the reader is referred to Koml´os and Simonovits 39 ).
As is typical when using the regularity lemma, the constant c(∆) obtained by Chv´atal et al.^38 was rather large (more precisely, it had the form of an exponential tower of 2’s of height ∆). More recently, Eaton^40 used a variant of the regularity lemma to show that one can take
c(∆) < 22
c∆
for some c > 0. Subsequently, Graham, R¨odl and Ruci´nski^41 showed that it is enough to take c(∆) < 2 c∆(log ∆)
2
for some c > 0 and ∆ > 1. They also show that there are graphs G with n vertices and maximum degree ∆ for which r(G) ≥ c∆ 0 n for some c 0 > 1 and n sufficiently large.
Chen and Schelp 42 extended the result by Chv´atal et al.^38 replacing the bounded degree condition by the following weaker requirement. A graph is said to be c-arrangeable if the vertices can be ordered, say, v 1 ,... , vn, such that for each i, |{j : vi ∼ vk, for k > i, and vk ∼ vj for j ≤ i}| ≤ c.
check the definition Chen and Schelp proved that for a fixed c, the Ramsey number for c-arrangeable graphs grows linearly with the size of the graph. They showed that a planar graph
(^38) V. Chv´atal, V. R¨odl, E. Szemer´edi and W. T. Trotter, The Ramsey number of a graph with bounded maximum degree, J. Comb. Theory Ser. B 34 (1983), 239–243. (^39) J. Koml´os and M. Simonovits, Szemer´edi’s regularity lemma and its applications in graph theory, Combinatorics, Paul Erd˝os is Eighty, Vol. 2, (D. Mikl´os, V. T. S´os, T. Sz˝onyi, eds.), Bolyai Soc. Math. Studies, 2 (1996), 97–132. (^40) N. Eaton, Ramsey numbers for sparse graphs, Discrete Math., to appear (^41) R. L. Graham, V. R¨odl and A. Ruci´nski, On graphs with linear Ramsey numbers, preprint. (^42) G. Chen and R. H. Schelp, Graphs with linearly bounded Ramsey numbers, J. Comb. Theory Ser. B 57 (1993), 138–149.
2.4. GRAPH RAMSEY THEORY 15
is 761-arrangeable, which was later improved to 10-arrangeable by Kierstead and Trotter^43 So, their results imply that planar graphs have linear Ramsey numbers.
Recently, R¨odl and Thomas^44 , generalizing results in 42 , showed that graphs with bounded genus have linear Ramsey numbers. The following three problems are in fact equivalent (subject to different constants).
Conjecture on Ramsey numbers for subgraphs with bounded average degrees (proposed by Burr and Erd˝os^37 ) For every graph G on n vertices in which every subgraph has average degree at most c, r(G) ≤ c′n where the constant c′^ depends only on c.
Conjecture on Ramsey numbers for bounded arboricity (proposed by Burr and Erd˝os^37 ) If a graph G on n vertices is the union of c forests, then r(G) ≤ c′n where the constant c′^ depends only on c.
Conjecture on Ramsey numbers for graphs with degree constraints (proposed by Burr and Erd˝os^37 ) For every graph G on n vertices in which every subgraph has minimum degree at most c, r(G) ≤ c′n where the constant c′^ depends only on c.
2.4.2. On relating graph Ramsey numbers to the classical Ramsey problems. The following several problems run along the lines of attempting to clarify the relationship between graph Ramsey numbers and the classical ones. Although these problems 45 46^ were raised very early on, little progress has been made so far.
(^43) H. A. Kierstead and W. T. Trotter, Planar graph colorings with an uncooperative partner, J. Graph Theory 18 (1994), 569-584. (^44) V. R¨odl and R. Thomas, Arrangeability and clique subdivisions, The Mathematics of Paul Erd˝os, II (R. L. Graham and J. Neˇsetˇril, eds.), 236-239, Springer-Verlag, Berlin, 1996. (^45) P. Erd˝os and R. L. Graham, On partition theorems for finite graphs, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erd˝os on his 60th birthday), Vol. I; Colloq. Math. Soc. J´anos Bolyai, Vol. 10, 515–527, North-Holland, Amsterdam, 1975. (^46) P. Erd˝os, On some problems in graph theory, combinatorial analysis and combinatorial number theory, Graph theory and combinatorics (Cambridge, 1983), 1–17, Academic Press, London-New York, 1984.
2.4. GRAPH RAMSEY THEORY 17
Conjecture (proposed by Burr and Erd˝os 49 ) For any tree T on n vertices, r(T ) ≤ 2 n − 2.
Clearly, for a star on n vertices, equality holds. So, the above conjecture can be restated as r(T ) ≤ r(Sn) where Sn denotes the star on n vertices.
The above problem is closely related to a conjecture by Erd˝os and S´os which will be discussed later in the chapter on extremal graph problems. This conjecture asserts that every graph with m vertices and more than (n − 2)m/2 edges contains every tree T on n vertices. If this conjecture were true, it would imply the above conjecture.
Suppose that a tree T has a 2-coloring with k vertices in one color and l vertices in the other. It was proved in 50 that
r(T ) ≥ max{ 2 k + l − 1 , 2 l − 1 }.
This leads to the following:
Problem 50 Is r(T ) = 4k for every tree which is a bipartite graph with k vertices in one color and 2k vertices in the other?
Chv´atal 51 proved that r(T, Km) = (m − 1)(n − 1) + 1
for any tree on n vertices. This result was generalized to graphs with small chro- matic number. For a graph G with chromatic number χ(G), it was shown 52 that
r(T, G) = (χ(G) − 1)(n − 1) + 1
for any tree T on n vertices, provided n is sufficiently large.
Conjecture 53 If m 1 ≤... ≤ mk, then r(T, Km 1 ,...,mk ) ≤ (χ(G) − 1)(r(T, Km 1 ,m 2 ) − 1) + m 1 where T is any tree on n vertices, and n is large enough.
2.4.4. On Ramsey numbers involving cycles.
Conjecture 54 For some > 0, r(C 4 , Kn) = o(n^2 −).
(^50) P. Erd˝os, R. Faudree, C. C. Rousseau and R. H. Schelp, Ramsey numbers for brooms, Pro- ceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982), Congr. Numer. 35 (1982), 283–293. (^51) V. Chv´atal, Tree-complete graph Ramsey numbers, J. Graph Theory 1 (1977), 93. (^52) S. A. Burr, P. Erd˝os, R. J. Faudree, R. J. Gould, M. S. Jacobson, C. C. Rousseau and R. H. Schelp, Goodness of trees for generalized books, Graphs Combin. 3 (1987) no. 1, 1–6.
18 2. RAMSEY THEORY
It is known that c(
n log n
)^2 > r(C 4 , Kn) > c(
n log n
)^3 /^2
where the lower bound is proved by probabilistic methods^33 , and the upper bound is due to Szemer´edi (unpublished^55 ).
For k fixed and n large, the probabilistic method gives r(Ck , Kn) > c(n/ log n)(k−1)/(k−2).
For the upper bound, it is known 55 56^ that for even k, we have
r(Ck , Kn) ≤ ck(n/ log n)1+1/m
where m = k/ 2 − 1.
For Ck, with k large compared to n, the Ramsey number r(Ck , Kn) was ob- tained by Bondy and Erd˝os^57 :
r(Ck , Kn) = (k − 1)(n − 1) + 1
for k > n^2 − 2.
Erd˝os, Faudree, Rousseau, Schelp^55 proposed the following problems:
Problem: Is it true that r(Ck , Kn) = (k − 1)(n − 1) + 1 if k ≥ n > 3?
Problem: What is the smallest value of k such that r(Ck , Kn) = (k − 1)(n − 1) + 1?
Problem: For a fixed n, what is the minimum value of r(Ck , Kn) over all k?
Together with Burr 58 , they proposed the following problem:
Problem Determine r(C 4 , K 1 ,n).
It is known that n + d
ne + 1 ≥ r(C 4 , K 1 ,n) ≥ n +
n − 6 n^11 /^40
(^55) P. Erd˝os, R. J. Faudree, C. C. Rousseau and R. H. Schelp, On cycle–complete graph Ramsey numbers, J. Graph Theory 2 (1978), 53–64. (^56) N. Alon, Independence numbers of locally sparse graphs and a Ramsey type problem, Random Structures and Algorithms 9 (1996), 271-278. (^57) J. A. Bondy and P. Erd˝os, Ramsey numbers for cycles in graphs, J. Combinatorial Theory Ser. B 14 (1973), 46–54. (^58) S. A. Burr, P. Erd˝os, R. J. Faudree, C. C. Rousseau and R. H. Schelp, Some complete bipartite graph–tree Ramsey numbers, Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), Ann. Discrete Math., 41, 79–89, North-Holland, Amsterdam-New York, 1989.
20 2. RAMSEY THEORY
can then derive:
r(3,... , 3 ︸ ︷︷ ︸ k
) − 1 ≤ 1 + k( r(3,... , 3 ︸ ︷︷ ︸ k− 1
≤ k!(
k!
(k − 1)!
r(3, 3 , 3 , 3) − 1 4!
< k!(e −
for k ≥ 4.
The lower bound for r(3,... , 3 ︸ ︷︷ ︸ k
) is closely related to the Schur number sk. A
subset of numbers S is said to be sum-free if whenever i and j are (not necessarily distinct) numbers in S then i + j is not in S. The Schur number sk is the largest integer such that numbers from 1 to sk can be partitioned into k sum-free sets. It can be shown 64 that, for k ≥ l,
r(3,... , 3 ︸ ︷︷ ︸ k
) − 2 ≥ sk ≥ c(2sl + 1)k/l
for some constant c.
Using a result of Exoo 65 giving s 5 ≥ 160, this implies
r(3,... , 3 ︸ ︷︷ ︸ k
) ≥ c(321)k/^5.
Conjecture ($250, a very old problem of Erd˝os’)
Determine lim k→∞ (r(3,... , 3 ︸ ︷︷ ︸ k
))^1 /k.
It is known (see 61 ) that r(3,... , 3 ︸ ︷︷ ︸ k
) is supermultiplicative in k so that the above
limit exists.
Problem ($100) Is the above limit finite or not?
Any improvement for small values of k will give a better general lower bound. The current range for this limit is between (321)^1 /^5 ≈ 3. 171765... and infinity.
(^64) F. R. K. Chung and C. M. Grinstead, A survey of bounds for classical Ramsey numbers, Journal of Graph Theory 7 (1983), 25-37. (^65) G. Exoo, A lower bound for Schur numbers and multicolor Ramsey numbers, Electronic J. of Combinatorics 1 (1994), # R8.
2.5. MULTI-COLORED RAMSEY NUMBERS 21
A conjecture on the ratio of multi-Ramsey numbers and Ramsey numbers (Proposed by Erd˝os and S´os^66 )
r(3, 3 , n) r(3, n)
as n → ∞.
Erd˝os^66 said, “It is very surprising that this problem which seems trivial at first sight should cause serious difficulties. We further expect that
r(3, 3 , n) n^2
as n → ∞ and perhaps
r(3, 3 , n) > n^3 −
for every > 0 if n is sufficiently large.”
A multi-colored Ramsey problem for odd cycles (proposed by Erd˝os and Graham 36 )
Show that for n ≥ 2 and any k,
lim k→∞
r(
k ︷ ︸︸ ︷ C 2 n+1,... , C 2 n+1) r(3,... , 3 ︸ ︷︷ ︸ k
This problem is open even for n = 2.
A multi-colored Ramsey problem for even cycles (proposed by Erd˝os and Graham 36 ) Determine r(C 2 m ,... , C 2 m ︸ ︷︷ ︸ k
It was proved 67 that
r(C 4 ,... , C 4 ︸ ︷︷ ︸ k
) ≤ k^2 + k + 1 for all k
r(C 4 ,... , C 4 ︸ ︷︷ ︸ k
) > k^2 − k + 1 for prime power k.
The following upper and lower bounds for r(C 2 m ,... , C 2 m ︸ ︷︷ ︸ k
) were given in 36 :
(^66) P. Erd˝os and V. T. S´os, Problems and results on Ramsey-Tur´an type theorems (prelim- inary report), Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, Calif., 1979), Congress. Numer. XXVI, pp. 17–23, Utilitas Math., Winnipeg, Man., 1980. (^67) F. R. K. Chung and R. L. Graham, On multicolor Ramsey numbers for complete bipartite graphs, J. Comb. Th. (B) 18 (1975), 164-69.
2.6. SIZE RAMSEY NUMBERS 23
For r(G 1 , G 2 ,... , Gk), some exact results are known when k ≤ 3 and the Gi’s are cycles, and for the case that G 1 is a large cycle and the others G’s are either odd cycles or complete subgraphs 55.
2.6. Size Ramsey numbers
The size Ramsey number ˆr(G, H) is the least integer m for which there exists a graph F with m edges so that in any coloring of the edges of F in red and blue, there is always either a red copy of G or a blue copy of H. Sometimes we write F → (G, H) to denote this. For G = H, we denote ˆr(G, G) by ˆr(G).
A size Ramsey problem for bounded degree graphs (proposed by Beck and Erd˝os 53 )
For a graph G on n vertices with bounded degree d , prove that ˆr(G) ≤ cn where c depends only on d.
The case for paths was proved by Beck 69 (also see 70 ) by using the following very nice result of P´osa 71 : Suppose that in a graph G, any subset X of the vertex set of size at most n satisfies:
|{y 6 ∈ X : y ∼ x ∈ X}| ≥ 2 |X| − 1.
Then G contains a path with 3n − 2 vertices.
Based on this result, Alon and Chung 72 explicitly construct a graph with cn edges so that no matter how we delete all but an -fraction of the vertices or edges, the remaining graph still contains a path of length n.
We point out that a directed version of this problem was considered by Erd˝os, Graham and Szemer´edi^73 in 1975. Let g(n) denote the least integer such that there is a directed acyclic graph G with g(n) edges having the property that for any set X of n vertices of G, there is a directed path on G of length n which does not hit X. Then they show
c 1
n log n log log n < g(n) < c 2 n log n
for constants c 1 , c 2 > 0.
(^69) J. Beck, On size Ramsey number of paths, trees, and circuits, I, J. Graph Theory 7 (1983), 115–129. (^70) J. Neˇsetˇril and V. R¨odl, eds., Mathematics of Ramsey Theory, Springer-Verlag, Berlin,
(^71) L. P´osa, Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), 359-364. (^72) N. Alon and F. R. K. Chung, Explicit constructions of linear-sized tolerant networks, Discrete Math. 72 (1988), 15-20. (^73) P. Erd˝os, R. L. Graham and E. Szemer´edi, On sparse graphs with dense long paths, Com- puters and mathematics with applications, pp. 365–369, Pergamon, Oxford, 1976
24 2. RAMSEY THEORY
Friedman and Pippenger 74 extended P´osa’s result:
Suppose that in a graph G, any subset X consisting of at most 2n − 2 vertices satisfies:
|{y 6 ∈ X : y ∼ x ∈ X}| ≥ (d + 1)|X|.
Then G contains every tree with n vertices and maximum degree at most d.
Using the above fact, they showed that
rˆ(T ) ≤ cn
for any tree with n vertices and bounded maximum degree.
Haxell, Kohayakawa, and Luczak 75 proved that the size Ramsey number for Cn has a linear upper bound.
For the complete graph Kn, Erd˝os, Faudree, Rousseau and Schelp 50 proved that
rˆ(Kn) =
r(n) 2
They asked the following size Ramsey problem for Kn,n:
Problem Determine ˆr(Kn,n).
Erd˝os, Faudree, Rousseau and Schelp 76 , and Neˇsetˇril and R¨odl 77 proved the following upper bound for ˆr(Kn,n).
ˆr(Kn,n) <
n^32 n.
For the lower bound, Erd˝os and Rousseau 78 proved by probabilistic methods that for n ≥ 6,
rˆ(Kn,n) >
n^22 n.
(^74) J. Friedman and N. Pippenger, Expanding graphs contain all small trees, Combinatorica 7 (1987), 71–76. (^75) P. E. Haxell, Y. Kohayakawa and T. Luczak, The induced size-Ramsey number of cycles, Combin. Probab. Comput. 4 (1995), 217–239. (^76) P. Erd˝os, R. J. Faudree, C. C. Rousseau and R. H. Schelp, The size Ramsey number, Period. Math. Hungar. 9 (1978), 145–161. (^77) J. Neˇsetˇril and V. R¨odl, The structure of critical graphs, Acta. Math. cad. Sci. Hungar. 32 (1978), 295-300. (^78) P. Erd˝os and C. C. Rousseau The size Ramsey number of a complete bipartite graph, Discrete Math. 113 (1993), 259-262.
