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41
INDEPENDENT SETS IN HYPERGRAPHS
, 2012
"... Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the ErdősStone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important t ..."
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Cited by 19 (2 self)
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Many important theorems and conjectures in combinatorics, such as the theorem of Szemerédi on arithmetic progressions and the ErdősStone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the socalled ‘sparse random setting’. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who developed general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural ‘counting ’ counterparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the wellknown conjecture of Kohayakawa, Luczak, and Rödl, a probabilistic embedding lemma for sparse graphs, for all 2balanced graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, such as sparse random versions of Szemerédi’s theorem, the ErdősStone Theorem and the ErdősSimonovits Stability Theorem, and obtain their natural ‘counting ’ versions, which in some cases are considerably stronger. We also obtain new results, such as a sparse version of the ErdősFranklRödl Theorem on the number of Hfree graphs and, as a consequence of the KLR conjecture, we extend a result of Rödl and Ruciński on Ramsey properties in sparse random graphs to the general, nonsymmetric setting. Similar results have been discovered independently by Saxton and Thomason.
Stability results for random discrete structures, Random Structures Algorithms
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The typical structure of sparse Kr+1free graphs
, 2013
"... Two central topics of study in combinatorics are the socalled evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of Hfree graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem tha ..."
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Cited by 8 (3 self)
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Two central topics of study in combinatorics are the socalled evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of Hfree graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical Hfree graph with n vertices and m edges changes as m grows from 0 to ex(n,H). In this paper, we resolve this problem in the case when H is a clique, extending a classical result of Kolaitis, Prömel, and Rothschild. In particular, we prove that for every r � 2, there is 2 − 2 an explicit constant θr such that, letting mr = θrn r+2(logn) 1/[(r+1 2)−1], the following holds for every positive constant ε. If m � (1 + ε)mr, then almost all Kr+1free nvertex graphs with m edges are rpartite, whereas if n ≪ m � (1−ε)mr, then almost all of them are not rpartite.
The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers
, 2012
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COUNTING SUMFREE SETS IN ABELIAN GROUPS
"... In this paper we study sumfree sets of order m in finite Abelian groups. We prove a general theorem on 3uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and as ..."
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Cited by 5 (2 self)
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In this paper we study sumfree sets of order m in finite Abelian groups. We prove a general theorem on 3uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sumfree sets of order m in Abelian groups G whose order is divisible by a prime q with q ≡ 2 (mod 3), for every m � C(q) √ n log n, thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sumfree subsets of size m are contained in a maximumsize sumfree subset of G. We also give a completely selfcontained proof of this statement for Abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of size m in an (n, d, λ)graph.