Abstract. Asymptotic extremal combinatorics deals with questions that in the language of model theory can be re-stated 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 sub-model 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 set of the form C " Z d , where C ` R d is convex and Z d denotes the integer lattice, is called a convex lattice set. It is known that the Helly number of d-dimensional convex lattice sets is 2 d . We prove that the fractional Helly number is only d + 1: for every d and every ff 2 (0; 1] there exists fi ? 0 such that whenever F 1 ; : : : ; Fn are convex lattice sets in Z d such that T i2I F i 6= ; for at least ff \Gamma n d+1 \Delta index sets I ` f1; 2; : : : ; ng of size d + 1 then there exists a (lattice) point common to at least fin of the F i . This implies a (p; d + 1)-theorem for every p d + 1; that is, if F is a finite family of convex lattice sets in Z d such that among every p sets of F , some d + 1 intersect, then F has a transversal of size bounded by a function of d and p. 1

Let be a triple system on an n element set. Suppose that contains more than (1/3 #) triples, where #>10 -6 is explicitly defined and n is su#ciently large. Then there is a set of four points containing at least three triples of F.This improves previous bounds of de Caen [1] and Matthias [7] .

#This author is partially supported by NSF grant DMS-0457512. 1 Also, we observe that the demonstrated non-principality phenomenon holds also with respect to the Ramsey-Tur'an density as well. 1 Introduction In this paper, we prove the non-principality phenomenon for the classical extremal problems for k-uniform 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 Ramsey-Tur'an theory, introduced by Erd""os and S'os. Although we prove the non-principality phenomenon for Ramsey-Tur'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 k-uniform hypergraphs (k-graphs for short), let

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 r-graph F with vertex set X is a family of r-element 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 r-graphs F, the Turán number ex(n, F) is the maximum number of edges in an n vertex r-graph 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 r-graphs F (see [3] for a survey of results). Determining the Turán density of complete r-graphs is a fundamental question about set-systems. 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-

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x = 1 2, but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.

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 (fractal-like) structure. We also give some necessary conditions for forcibility, which imply that finitely forcible graphons are “rare”, and exhibit simple and explicit non-forcible graphons.

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 called TURÁN TYPE EXTREMAL PROBLEM.

Given positive integers n, k, t, with 2 ≤ k ≤ n, and t < 2k, let m(n, k, t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k − 1)-subset of [n] contains at most t − 1 members of F. For fixed k and t, we determine the order of magnitude of m(n, k, t). We also consider related Turán numbers T≥r(n, k, t) and Tr(n, k, t), where T≥r(n, k, t) (Tr(n, k, t)) denotes the minimum size of a family F ⊂ � � � � [n]

For an r-graph 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 r-graphs, the co-degree Turán number co-ex(n, F) is the maximum of C(H) among all r-graphs H which contain no member of F as a subhypergraph. Define the co-degree density of a family F to be γ(F) = lim sup n→∞ co-ex(n, F) n When r ≥ 3, non-zero values of γ(F) are known for very few finite r-graphs 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.