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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 74 (6 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
Finitely forcible graphons
 arXiv:0901.0929. BÉLA BOLLOBÁS, SVANTE JANSON, AND
, 2009
"... We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Tu ..."
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We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turán, Erdős–Simonovits and Chung–Graham–Wilson, we construct numerous finitely forcible graphons. Most of these fall into two categories: one type has an algebraic structure and the other type has an iterated (fractallike) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are “rare”, and exhibit simple and explicit nonforcible graphons.
Asymptotic structure of graphs with the minimum number of triangles
, 2012
"... We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density. ..."
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Cited by 14 (2 self)
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We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.
On hypergraphs with every four points spanning at most two triples
 Electron. J. Combin
, 2003
"... ..."
Cubesupersaturated graphs and related problems
 IN PROGRESS IN GRAPH THEORY
, 1982
"... In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges. Given a graph L, ex (n, L) will denote the maximum number of edges a graph G " of order n can have without containing any L. Determining ex(n,L), or at least finding good bounds on it will be ..."
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Cited by 10 (1 self)
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In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges. Given a graph L, ex (n, L) will denote the maximum number of edges a graph G &quot; of order n can have without containing any L. Determining ex(n,L), or at least finding good bounds on it will be called TURÁN TYPE EXTREMAL PROBLEM.
The codegree density of the Fano plane
 J. Combin. Theory Ser. B
"... It is shown that every n vertex triple system with every pair of vertices lying in at least (1/2 + o(1))n triples contains a copy of the Fano plane. The constant 1/2 is sharp. This is the first triple system for which such a (nonzero) constant is determined. Let X be a finite set. An rgraph F with ..."
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It is shown that every n vertex triple system with every pair of vertices lying in at least (1/2 + o(1))n triples contains a copy of the Fano plane. The constant 1/2 is sharp. This is the first triple system for which such a (nonzero) constant is determined. Let X be a finite set. An rgraph F with vertex set X is a family of relement subsets of X. These subsets are called edges, and X is written as V (F). When there is no confusion, we denote an edge {x, y, z} as xyz. Given a family of rgraphs F, the Turán number ex(n, F) is the maximum number of edges in an n vertex rgraph containing no member of F. The Turán density of F is. If F = {F}, we write π(F) instead of π({F}). For r> 2, defined as π(F): = limn→ ∞ ex(n, F) / � n r computing π(F) when it is nonzero is notoriously hard, even for very simple rgraphs F (see [3] for a survey of results). Determining the Turán density of complete rgraphs is a fundamental question about setsystems. In fact, this is not known in any nontrivial case when r ≥ 3. In the past few years, there has been some progress on related problems. The Fano plane is the projective plane P G(2, 2) consisting of seven vertices (points) and seven edges (lines). Al
Codegree density of hypergraphs
 J. Combin. Theory (A
"... For an rgraph H, let C(H) = minS d(S), where the minimum is taken over all (r − 1)sets of vertices of H, and d(S) is the number of vertices v such that S ∪ {v} is an edge of H. Given a family F of rgraphs, the codegree Turán number coex(n,F) is the maximum of C(H) among all rgraphs H which c ..."
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Cited by 9 (3 self)
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For an rgraph H, let C(H) = minS d(S), where the minimum is taken over all (r − 1)sets of vertices of H, and d(S) is the number of vertices v such that S ∪ {v} is an edge of H. Given a family F of rgraphs, the codegree Turán number coex(n,F) is the maximum of C(H) among all rgraphs H which contain no member of F as a subhypergraph. Define the codegree density of a family F to be γ(F) = lim supn→∞ coex(n,F) n When r ≥ 3, nonzero values of γ(F) are known for very few finite rgraphs families F. Nevertheless, our main result implies that the possible values of γ(F) form a dense set in [0, 1). The corresponding problem in terms of the classical Turán density is an old question of Erdős (the jump constant conjecture), which was partially answered by Frankl and Rödl [14]. We also prove the existence, by explicit construction, of finite F satisfying 0 < γ(F) < minF∈F γ(F). This is parallel to recent results on the Turán density by Balogh [1], and by the first author and Pikhurko [23]. 1.
The Maximum Number of Disjoint Pairs in a Family of Subsets
 Graphs and Combinatorics
, 1985
"... Abstract. Let.~ " be a family of 2 "+1 subsets of a 2nelement set. Then the number of disjoint pairs in ~ " is bounded by (1 + o(1))22". This proves an old conjecture of Erd/Ss. Let ~ " be a family of 2 tl/tk+l~+~ " subsets of an nelement set. Then the number of conta ..."
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Cited by 8 (2 self)
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Abstract. Let.~ " be a family of 2 "+1 subsets of a 2nelement set. Then the number of disjoint pairs in ~ " is bounded by (1 + o(1))22". This proves an old conjecture of Erd/Ss. Let ~ " be a family of 2 tl/tk+l~+~ " subsets of an nelement set. Then the number of containments in, ~ is bounded by I~1 'X (1 1/k + o(1)) k 2: " This verifies a conjecture of Daykin and Erd/Ss. A similar Erd6sStone type result is proved for the maximum number of disjoint pairs in a family of subsets. I.
Constructions of NonPrincipal Families in Extremal Hypergraph Theory
"... #This author is partially supported by NSF grant DMS0457512. 1 Also, we observe that the demonstrated nonprincipality phenomenon holds also with respect to the RamseyTur'an density as well. 1 Introduction In this paper, we prove the nonprincipality phenomenon for the classical extremal prob ..."
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#This author is partially supported by NSF grant DMS0457512. 1 Also, we observe that the demonstrated nonprincipality phenomenon holds also with respect to the RamseyTur'an density as well. 1 Introduction In this paper, we prove the nonprincipality phenomenon for the classical extremal problems for kuniform hypergraphs. The main motivation is to study the qualitative difference between the cases k = 2, and k> = 3, and our results for the Tur'an problem exhibit this difference. We also study this question in the context of RamseyTur'an theory, introduced by Erd&quot;&quot;os and S'os. Although we prove the nonprincipality phenomenon for RamseyTur'an problems when k> = 3, the behavior for k = 2 remains open. This is one of the few cases where an extremal problem for hypergraphs can be solved but not for graphs. Given a family F of kuniform hypergraphs (kgraphs for short), let