Previous work on Ramsey multiplicity focused primarily on the multiplicity of complete graphs of constant size. We study the question for larger complete graphs, specifically showing that in every 2-coloring of a complete graph on @math vertices, there are at least @math monochromatic complete subgraphs of size between @math and @math . We also study bounds on the ratio between a maximum and a random red clique in a graph, and the ratio between a maximum and a random monochromatic clique in a graph.
In 1995, Shearer @cite_4 used the probabilistic method to prove that @math , where @math is the size of the maximum independent set and @math is the average degree in the graph. Following his technique, Alon @cite_0 proved that for a graph in which the neighborhood of every vertex is @math -colorable, @math for some constant @math . Note that an @math -colorable graph is @math -free, since a clique can contain at most one vertex of each color.
@cite_0 @cite_4
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We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of @math -dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS'05, SICOMP'08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP'07] concerning testing of matrix properties. The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that for any finite or infinite family @math of forbidden vertex-ordered graphs, and any @math , there exist @math and @math so that any vertex-ordered graph which is @math -far from being @math -free contains at least @math copies of some @math (with the correct vertex order) where @math . The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for @math -partite graphs with "undesirable" edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph. Oliveira conjectured that the order of the mixing time of the exclusion process with @math -particles on an arbitrary @math -vertex graph is at most that of the mixing-time of @math independent particles. We verify this up to a constant factor for @math -regular graphs when each edge rings at rate @math in various cases: (1) when @math or (2) when @math the spectral-gap of a single walk is @math and @math or (3) when @math for some constant @math . In these cases our analysis yields a probabilistic proof of a weaker version of Aldous' famous spectral-gap conjecture (resolved by ). We also prove a general bound which is at worst @math , which is within a @math factor from Oliveira's conjecture when @math . As applications we get new mixing bounds: (a) @math for expanders, (b) order @math for the hypercube @math and (c) order @math for vertex-transitive graphs of moderate growth and for supercritical percolation on a fixed dimensional torus. Indyk and Sidiropoulos (2007) proved that any orientable graph of genus @math can be probabilistically embedded into a graph of genus @math with constant distortion. Viewing a graph of genus @math as embedded on the surface of a sphere with @math handles attached, Indyk and Sidiropoulos' method gives an embedding into a distribution over planar graphs with distortion @math , by iteratively removing the handles. By removing all @math handles at once, we present a probabilistic embedding with distortion @math for both orientable and non-orientable graphs. Our result is obtained by showing that the nimum-cut graph of Erickson and Har Peled (2004) has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma of Lee and Sidiropoulos (2009). 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 Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998