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Flag algebras
 Journal of Symbolic Logic
"... Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a random ..."
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Cited by 80 (7 self)
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Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be restated as follows. For finite models M, N of an universal theory without constants and function symbols (like graphs, digraphs or hypergraphs), let p(M, N) be the probability that a randomly chosen submodel of N with M  elements is isomorphic to M. Which asymptotic relations exist between the quantities p(M1, N),..., p(Mh, N), where M1,..., Mh are fixed “template ” models and N  grows to infinity? In this paper we develop a formal calculus that captures many standard arguments in the area, both previously known and apparently new. We give the first application of this formalism by presenting a new simple proof of a result by Fisher about the minimal possible density of triangles in a graph with given edge density. §1. Introduction. A substantial part of modern extremal combinatorics (which will be called here asymptotic extremal combinatorics) studies densities with which some “template ” combinatorial structures may or may not appear in unknown (large) structures of the same type1. As a typical example, let Gn be a
A disproof of a conjecture of Erdős in Ramsey theory
 J. London Math. Soc
, 1989
"... Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of mono ..."
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Cited by 38 (0 self)
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Denote by kt(G) the number of complete subgraphs of order f in the graph G. Let where G denotes the complement of G and \G \ the number of vertices. A wellknown conjecture of Erdos, related to Ramsey's theorem, is that Mmn^K ct(ri) = 2 1 ~ { * ). This latter number is the proportion of monochromatic Kt's in a random colouring of Kn. We present counterexamples to this conjecture and discuss properties of the extremal graphs. 1.
On Perfect Matchings in Uniform Hypergraphs with . . .
, 2009
"... We study sufficient ℓdegree (1 ≤ ℓ < k) conditions for the appearance of perfect and nearly perfect matchings in kuniform hypergraphs. In particular, we obtain a minimum vertex degree condition (ℓ = 1) for 3uniform hypergraphs, which is approximately tight, by showing that every 3uniform hyp ..."
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Cited by 33 (4 self)
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We study sufficient ℓdegree (1 ≤ ℓ < k) conditions for the appearance of perfect and nearly perfect matchings in kuniform hypergraphs. In particular, we obtain a minimum vertex degree condition (ℓ = 1) for 3uniform hypergraphs, which is approximately tight, by showing that every 3uniform hypergraph on n vertices with minimum vertex degree at least (5/9+o(1)) `n´ 2 contains a perfect matching.
Extremal results in sparse pseudorandom graphs
 Adv. Math. 256 (2014), 206–290. arXiv:1204.6645 doi:10.1016/j.aim.2013.12.004 MR3177293
"... 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 (9 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
A geometric theory for hypergraph matching
 Memoirs of the American Mathematical Society
, 1908
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Undecidability of linear inequalities in graph homomorphism densities
 Journal of the American Mathematical Society
"... Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic in ..."
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Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. As is explained below, for dense graphs, it is possible to replace subgraph densities with homomorphism densities. An easy observation shows that one can convert any algebraic inequality between
SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
 KRAGUJEVAC J. MATH. 25 (2003) 31–49.
, 2003
"... Let G = (V, E) be a simple graph with n vertices, e edges, and vertex degrees d1, d2,..., dn. Let d1, dn be the highest and the lowest degree of vertices of G and mi be the average of the degrees of the vertices adjacent to vi ∈ V. We prove that n� i=1 d 2 � � 2e i = e + n − 2 n − 1 if and only if ..."
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Cited by 9 (0 self)
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Let G = (V, E) be a simple graph with n vertices, e edges, and vertex degrees d1, d2,..., dn. Let d1, dn be the highest and the lowest degree of vertices of G and mi be the average of the degrees of the vertices adjacent to vi ∈ V. We prove that n� i=1 d 2 � � 2e i = e + n − 2 n − 1 if and only if G is a star graph or a complete graph or a complete graph with one isolated vertex. We establish the following upper bound for the sum of the squares of the degrees of a graph G: n� i=1 d 2 � �
Asymptotic quantization of exponential random graphs
, 2013
"... Abstract. We describe the asymptotic properties of the edgetriangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from ..."
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Abstract. We describe the asymptotic properties of the edgetriangle exponential random graph model as the natural parameters diverge along straight lines. We show that as we continuously vary the slopes of these lines, a typical graph drawn from this model exhibits quantized behavior, jumping from one complete multipartite graph to another, and the jumps happen precisely at the normal lines of a polyhedral set with infinitely many facets. As a result, we provide a complete description of all asymptotic extremal behaviors of the model. 1.