460 research outputs found

    A contribution to the Zarankiewicz problem

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    Given positive integers m,n,s,t, let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m-by-n that does not contain an all ones submatrix of size s-by-t. We find a flexible upper bound on z(m,n,s,t) that implies the known bounds of Kovari, Sos and Turan, and of Furedi. As a consequence, we find an upper bound on the spectral radius of a graph of order n without a complete bipartite subgraph K_{s,t}

    Density version of the Ramsey problem and the directed Ramsey problem

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    We discuss a variant of the Ramsey and the directed Ramsey problem. First, consider a complete graph on nn vertices and a two-coloring of the edges such that every edge is colored with at least one color and the number of bicolored edges ERB|E_{RB}| is given. The aim is to find the maximal size ff of a monochromatic clique which is guaranteed by such a coloring. Analogously, in the second problem we consider semicomplete digraph on nn vertices such that the number of bi-oriented edges Ebi|E_{bi}| is given. The aim is to bound the size FF of the maximal transitive subtournament that is guaranteed by such a digraph. Applying probabilistic and analytic tools and constructive methods we show that if ERB=Ebi=p(n2)|E_{RB}|=|E_{bi}| = p{n\choose 2}, (p[0,1)p\in [0,1)), then f,F<Cplog(n)f, F < C_p\log(n) where CpC_p only depend on pp, while if m=(n2)ERB<n3/2m={n \choose 2} - |E_{RB}| <n^{3/2} then f=Θ(n2m+n)f= \Theta (\frac{n^2}{m+n}). The latter case is strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| = p{n\choose 2}

    Dense H-free graphs are almost (Χ(H)-1)-partite

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    By using the Szemeredi Regularity Lemma, Alon and Sudakov recently extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε>0 and sufficiently large n the following is true. Whenever G is an n-vertex graph with minimum degree δ(G)≥(1 − 3/3r−1 + ε)n, either G contains H, or we can delete f(n,H)≤Cn2−1/t edges from G to obtain an r-partite graph. Further, we are able to determine the correct order of magnitude of f(n,H) in terms of the Zarankiewicz extremal function

    The history of degenerate (bipartite) extremal graph problems

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    This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

    Some remarks on the Zarankiewicz problem

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    The Zarankiewicz problem asks for an estimate on z(m,n;s,t), the largest number of 1's in an m×n matrix with all entries 0 or 1 containing no s×t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m,n;s,t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method
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