his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_

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Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number: provided g(G)> 4. 1

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

For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, and for maximum distance separable (MDS) codes specifically, and show how they improve upon previous results. For MDS codes, the new bounds are found by upper-bounding the stopping redundancy by a combinatorial quantity closely related to Turán numbers. (The Turán number, „

In this paper we obtain an asymptotic generalization of Turán's theorem. We prove that if all the non-trivial eigenvalues of a d-regular graph G on n vertices are sufficiently small, then the largest K t -free subgraph of G contains approximately t 1 -fraction of its edges. Turán's theorem corresponds to the case d = n 1.

We prove the analogue of Turán’s Theorem in random graphs with edge probability p(n) ≫ n−1/(k−1.5) 1. With probability 1 − o(1), one needs to delete approximately k−1 –fraction of the edges in a random graph in order to destroy all cliques of size k. 1

We consider embeddings of the complete t-ary trees of depth k (denotation T k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T k,t), is known if max{k, t} ≤ 2. First we study the next open case max{k, t} = 3. We improve the known upper bound dim(T k,3) ≤ 2k + 1 up to limk→ ∞ dim(T k,3)/k ≤ 5/3 and derive the asymptotic limt→ ∞ dim(T 3,t)/t = 227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T k,t) for arbitrary k and t.

In 1941, Turán conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turán (3, 4)-graph) is ≥ 4/9(1− o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. Fon-der-Flaass (1988) presented a general construction that converts an arbitrary ⃗ C4-free orgraph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the bound ≥ 3/7(1 − o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound 4/9(1 − o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-der-Flaass construction under any of the following assumptions on the undirected graph G underlying the orgraph Γ: • G is complete multipartite; • the edge density of G is ≥ (2/3 − ɛ) for some absolute constant ɛ> 0. We are also able to improve Fon-der-Flaass’s bound to 7/16(1 − o(1)) without any extra assumptions on Γ.

The minimal number of triples required to represent all quintuples on an n-element set is determined for n 13 and all extremal constructions are found. In particular we establish that there is a unique minimal system on 13 points, namely the 52 collinear triples of the projective plane of order 3. 1 Introduction. We say that a set T represents another set F if T is contained in F . A (n; k; r)- Tur'an system is a pair (X ; B) where B is a collection of r-tuples of the n-element set X such every k-element subset of X is represented by some member of B. The size of a (n; k; r)-Tur'an system (X ; B) is the number of r-tuples in B and T (n; k; r) is the minimum size required for a (n; k; r)-Tur'an system to exist. Partialy supported by NSF grant CCR-8920692. The problem of determining T (n; k; r) was solved by Tur'an [12] for the case r = 2: T (n; s + 1; 2) = mn \Gamma m(m+ 1) 2 \Delta s with m n s m+ 1 The general problem also was formulated by Tur'an and was first publis...