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48
Testing Subgraphs in Directed Graphs
 Proc. of the 35 th Annual Symp. on Theory of Computing (STOC
, 2003
"... Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Sz ..."
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Cited by 62 (15 self)
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Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least #n edges have to be deleted from it to make it Hfree. We show that in this case G contains at least f(#, H)n copies of H. This is proved by establishing a directed version of Szemeredi's regularity lemma, and implies that for every H there is a onesided error property tester whose query complexity is bounded by a function of # only for testing the property PH of being Hfree.
Large K_rfree subgraphs in K_sfree graphs and some other Ramseytype problems
 RANDOM STRUCTURES & ALGORITHMS
, 2005
"... In this paper we present three Ramseytype results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rog ..."
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Cited by 22 (10 self)
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In this paper we present three Ramseytype results, which we derive from a simple and yet powerful lemma, proved using probabilistic arguments. Let 3 ≤ r < s be fixed integers and let G be a graph on n vertices not containing a complete graph Ks on s vertices. More than 40 years ago Erdős and Rogers posed the problem of estimating the maximum size of a subset of G without a copy of the complete graph Kr. Our first result provides a new lower bound for this problem, which improves previous results of various researchers. It also allows us to solve some special cases of a closely related question posed by Erdős. For two graphs G and H, the Ramsey number R(G, H) is the minimum integer N such that any redblue coloring of the edges of the complete graph KN, contains either a red copy of G or a blue copy of H. The book with n pages is the graph Bn consisting of n triangles sharing one edge. Here we study the bookcomplete graph Ramsey numbers and show that R(Bn, Kn) ≤ O(n 3 / log 3/2 n), improving the bound of Li and Rousseau. Finally, motivated by a question of Erdős, Hajnal, Simonovits, Sós, and Szemerédi, we obtain for all 0 <δ<2/3 an estimate on the number of edges in a K4free graph of order n which has no independent set of size
THE STRUCTURE OF ALMOST ALL GRAPHS IN A HEREDITARY PROPERTY
"... A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n ↦ → Pn, where Pn denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then ..."
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Cited by 18 (8 self)
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A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n ↦ → Pn, where Pn denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then Pn  = 2 (1−1/r+o(1))n2 /2 where r = r(P) ∈ N is the socalled ‘colouring number ’ of P. However, their results tell us very little about the structure of a typical graph G ∈ P. In this paper we describe the structure of almost every graph in a hereditary property of graphs, P. As a consequence, we derive essentially optimal bounds on the speed of P, improving the AlekseevBollobásThomason Theorem, and also generalizing results of Balogh, Bollobás and Simonovits.
Extremal results in sparse pseudorandom graphs
 ADV. MATH. 256 (2014), 206–290
, 2014
"... 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 extendin ..."
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Cited by 13 (8 self)
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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 wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s