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We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2) and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kučera. In this paper we present an efficient algorithm for all k > cn 0.5 , for any fixed c > 0, thus improving the trivial case k > cn 0.5 (log n) 0.5. The algorithm is based on the spectral properties of the graph.

For a graph H and an integer n, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H. H is r-degenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, . This is tight for all values of r and can also be derived from an earlier result of Füredi. We also show that there is an absolute positive constant c such that, for every fixed bipartite r-degenerate graph H, This is motivated by a conjecture of Erdős that asserts that, for every such H, For two graphs G and H, the Ramsey number is the minimum number n such that, in any colouring of the edges of the complete graph on n vertices by red and blue, there is either a red copy of G or a blue copy of H. Erdős conjectured that there is an absolute constant c such that, for any graph G with m edges, . Here we prove this conjecture for bipartite graphs G, and prove that for general graphs G with m edges, for some absolute positive constant c.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rödl, Kostochka, Gowers and Sudakov.

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truly random graphs. He also supplied several examples of pseudo-random graphs and discussed many of their properties. Thomason's papers undoubtedly defined directions of future research for many years.Another cornerstone contribution belongs to Chung, Graham and Wilson [26] who in 1989 showed that many properties of different nature are in certain sense equivalent to the notion of pseudo-randomness, defined using the edge distribution. This fundamental result opened many new horizons by showing additional facets of pseudo-randomness.Last years brought many new and striking results on pseudo-randomness by various researchers. There are two clear trends in recent research on pseudo-random graphs. The first is to apply very diverse methods from different fields (algebraic, linear algebraic, combinatorial, probabilistic etc.) to construct and study pseudo-random graphs. The second and equally encouraging is to find applications, in many cases quite surprising, of pseudo-random graphs to problems in Graph Theory, Computer Science and other disciplines. This mutually enriching interplay has greatly contributed to significant progress in research on pseudo-randomness achieved lately.The aim of this survey is to provide a systematic treatment of the concept of pseudo-random graphs, probably the first since the two seminal contributions of Thomason [79], [80]. Research in pseudo-random graphs has developed tremendously since then, making it impossible to provide full coverage of this subject in a single paper. We are thus forced to omit quite a few directions, approaches, theorem proofs from our discussion. Nevertheless we will attempt to provide the reader with a rather detailed and illustrative account of the current state of research in pseudo-random graphs.Although, as we will discuss later, there are several possible formal approaches to pseudorandomness, we will mostly emphasize the approach based on graph eigenvalues. We find this approach, combining linear algebraic and combinatorial tools in a very elegant way, probably the most appealing, convenient and yet quite powerful.This survey is structured as follows. In the next section we will discuss various formal definitions of the notion of pseudo-randomness, from the so called jumbled graphs of Thomason to the (n, d, λ)-graphs defined by Alon, where pseudo-randomness is connected to the eigenvalue gap. We then describe several known constructions of pseudo-random graphs, serving both as illustrative examples for the notion of pseudo-randomness, and also as test cases for many of the theorems to be presented afterwards. The strength of every abstract concept is best tested by properties it enables to derive. Pseudo-random graphs are certainly not an exception here, so in Section 4 we discuss various properties of pseudo-random graphs. Section 5, the final section of the paper, is devoted to concluding remarks. Definitions of pseudo-random graphsPseudo-random graphs are much more of a general concept describing some graph theoretic phenomenon than of a rigid wel...

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − )n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < < 1, there exists a constant c = c(d, ) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − )n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and , then G has a copy of every tree T as above.

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