Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rödl, Łuczak-Rödl, Prömel-Rödl, Erdős-Hajnal, and Nikiforov. The proofs are based on a simple lemma (generalizing one by Graham, Rödl, and Ruciński) that can be used as a replacement for Szemerédi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.     

Extremal results in sparse pseudorandom graphs

Szemerédi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs.     

Direct Ramsey theorem for structures involving relations and functions

We prove the direct structural Ramsey theorem for structures with relations as well as functions. The result extends the theorem of Abramson and Harrington and of Nešet?il and Rödl.     

A variant of the hypergraph removal lemma

Recent work of Gowers [T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001) 465-588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299-345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275-291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes.     

Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.     

On splittable colorings of graphs and hypergraphs

The notion of a split coloring of a complete graph was introduced by Erd?s and Gyárfás [ 7 ] as a generalization of split graphs. In this work, we offer an alternate interpretation by comparing such a coloring to the classical Ramsey coloring problem via a two‐round game played against an adversary. We show that the techniques used and bounds obtained on the extremal (r,m)‐split coloring problem of [ 7 ] are closer in nature to the Turán theory of graphs rather than Ramsey theory. We extend the notion of these colorings to hypergraphs and provide bounds and some exact results. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 226–237, 2002     

Ramsey numbers of sparse hypergraphs

Chvátal, Rödl, Szemerédi and Trotter [V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, The Ramsey number of a graph with a bounded maximum degree, J. Combinatorial Theory B 34 (1983), 239–243] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus, 3-uniform hypergraphs of bounded degree have linear Ramsey numbers, submitted] and [B. Nagle, S. Olsen, V. Rödl and M. Schacht, On the Ramsey number of sparse 3-graphs, preprint] the same result was proved for 3-uniform hypergraphs. In [O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus, Embeddings and Ramsey numbers of sparse k-uniform hypergraphs, submitted] we extended this result to k-uniform hypergraphs for any integer $k\ge 3$. As in the 3-uniform case, the main new tool which we proved and used is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform ‘quasi-random’ hypergraphs.     

Bounding Ramsey numbers through large deviation inequalities

We develop a new approach for proving lower bounds for various Ramsey numbers, based on using large deviation inequalities. This approach enables us to obtain the bounds for the off-diagonal Ramsey numbers R(K^r, K^k), r ≤ k, that match the best known bounds, obtained through the local lemma. We discuss also the bounds for a related Ramsey-type problem and show, for example, that there exists a K⁴-free graph G on n vertices in which every cn^3/5 log^1/2 n vertices span a copy of K³. © 1995 John Wiley & Sons, Inc.     

Constructing Ramsey graphs from Boolean function representations

Explicit construction of Ramsey graphs or graphs with no large clique or independent set, has remained a challenging open problem for a long time. While Erdös’ probabilistic argument shows the existence of graphs on 2n vertices with no clique or independent set of size 2 ⁿ , the best explicit constructions achieve a far weaker bound. There is a connection between Ramsey graph constructions and polynomial representations of Boolean functions due to Grolmusz; a low degree representation for the OR function can be used to explicitly construct Ramsey graphs [17,18]. We generalize the above relation by proposing a new framework. We propose a new definition of OR representations: a pair of polynomials represent the OR function if the union of their zero sets contains all points in {0, 1} ⁿ except the origin. We give a simple construction of a Ramsey graph using such polynomials. Furthermore, we show that all the known algebraic constructions, ones to due to Frankl-Wilson [12], Grolmusz [18] and Alon [2] are captured by this framework; they can all be derived from various OR representations of degree O(√n) based on symmetric polynomials. Thus the barrier to better Ramsey constructions through such algebraic methods appears to be the construction of lower degree representations. Using new algebraic techniques, we show that better bounds cannot be obtained using symmetric polynomials.     

Note on the 3-graph counting lemma

Szemerédi's regularity lemma proved to be a powerful tool in extremal graph theory. Many of its applications are based on the so-called counting lemma: if G is a k-partite graph with k-partition V₁∪?∪V_k, |V₁|=?=|V_k|=n, where all induced bipartite graphs G[V_i,V_j] are (d,ε)-regular, then the number of k-cliques K_k in G is . Frankl and Rödl extended Szemerédi's regularity lemma to 3-graphs and Nagle and Rödl established an accompanying 3-graph counting lemma analogous to the graph counting lemma above. In this paper, we provide a new proof of the 3-graph counting lemma.     

用Paley图计算对角Ramsey数下界的新方法

本文研究了对角Paley数的下界问题.利用一个新发现的Paley图的自同构,给出了计算Paley图团数的一个新方法,获得了2个对角Rasey数的新下界:R(20,20)≥18877,R(21,21)≥25949.     

Linear Ramsey Numbers for Bounded-Degree Hypergrahps

We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), pp. 239–243] showed in 1983. Our proof demonstrates the potential of a new regularity lemma by [Y. Ishigami, A simple regularization of hypergraphs, preprint, arXiv:math/0612838 (2006)].     

Eigenvalue bounds for independent sets

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdős–Rényi graphs. We investigate further properties of our bounds, and show how our results on the Erdős–Rényi graphs can be extended to other polarity graphs.     

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements. Research supported by an NSF Graduate Research Fellowship and a Princeton Centennial Fellowship. Research supported in part by NSF CAREER award DMS-0812005 and by USA-Israeli BSF grant.     

Embedding theorems for classes of convex sets

Rådström's embedding theorem states that the nonempty compact convex subsets of a normed vector space can be identified with points of another normed vector space such that the embedding map is additive, positively homogeneous, and isometric. In the present paper, extensions of Rådström's embedding theorem are proven which provide additional information on the embedding space. These results include those of Hörmander who proved a similar embedding theorem for the nonempty closed bounded convex subsets of a Hausdorff locally convex vector space. In contrast to Hörmander's approach via support functionals, all embedding theorems of the present paper are proven by a refinement of Rådström's original method which is constructive and does not rely on Zorn's lemma. This paper also includes a brief discussion of some actual or potential applications of embedding theorems for classes of convex sets in probability theory, mathematical economics, interval mathematics, and related areas.     

On the KŁR conjecture in random graphs

The K?R conjecture of Kohayakawa, ?uczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph G _n,p , for sufficiently large p:= p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.     

On unavoidable‐induced subgraphs in large prime graphs

Chudnovsky, Kim, Oum, and Seymour recently established that any prime graph contains one of a short list of induced prime subgraphs [1]. In the present article, we reprove their theorem using many of the same ideas, but with the key model‐theoretic ingredient of first determining the so‐called amount of stability of the graph. This approach changes the applicable Ramsey theorem, improves the bounds and offers a different structural perspective on the graphs in question. Complementing this, we give an infinitary proof that implies the finite result.     

Conditions on Ramsey Nonequivalence

Given a graph H , a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G , H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. (J Combin Theory Ser B 109 (2014), 120–133) asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved that a clique is not Ramsey equivalent to any other connected graph. Results of Ne?et?il et al. showed that any two graphs with different clique number (Combinatorica 1(2) (1981), 199–202) or different odd girth (Comment Math Univ Carolin 20(3) (1979), 565–582) are not Ramsey equivalent. These are the only structural graph parameters we know that “distinguish” two graphs in the above sense. This article provides further supportive evidence for a negative answer to the question of Fox et al. by claiming that for wide classes of graphs, the chromatic number is a distinguishing parameter. In addition, it is shown here that all stars and paths and all connected graphs on at most five vertices are not Ramsey equivalent to any other connected graph. Moreover, two connected graphs are not Ramsey equivalent if they belong to a special class of trees or to classes of graphs with clique‐reduction properties.     

On the Ramsey Number of Sparse 3-Graphs

We consider a hypergraph generalization of a conjecture of Burr and Erd?s concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), no. 3, 239–243] that the Ramsey number R(G) of a graph G of bounded maximum degree is linear in |V(G)|. We derive the analogous result for 3-uniform hypergraphs.     

Almost all graphs are rigid—revisited

It is shown that almost all graphs are unretractive, i.e. have no endomorphisms other than their automorphisms. A more general result has already been published in [V. Koubek, V. Rödl, On the minimum order of graphs with given semigroup, J. Combin. Theory Ser. B 36 (1984) 135–155]. In the paper at hand, a different proof is presented, following an approach of P. Erdős and A. Rényi that was used in their proof [P. Erdős, A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963) 295–315] that almost all graphs are asymmetric (have a trivial automorphism group). The approach is modified using an algebraically motivated reduction to idempotent endomorphisms. These take the role of the automorphisms in the proof of Erdős and Rényi. A bound of is provided for the ratio of retractive graphs among all graphs with n vertices, confirming an earlier statement by Babai [L. Babai, Automorphism groups, isomorphism, reconstruction, in: R.L. Graham, M. Grötschel, L. Lovász (Eds.), in: Handbook of Combinatorics, vol. 2, Elsevier, Amsterdam, 1995, pp. 1447–1540]. The fact that almost all graphs are unretractive and asymmetric can be summarized in the statement that almost all graphs are rigid (have a trivial endomorphism monoid), and the same bound can be obtained for corresponding ratios of nonrigid graphs.