460 research outputs found
A contribution to the Zarankiewicz problem
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
We discuss a variant of the Ramsey and the directed Ramsey problem. First,
consider a complete graph on 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 is given. The aim is to find the maximal size of a
monochromatic clique which is guaranteed by such a coloring. Analogously, in
the second problem we consider semicomplete digraph on vertices such that
the number of bi-oriented edges is given. The aim is to bound the
size of the maximal transitive subtournament that is guaranteed by such a
digraph.
Applying probabilistic and analytic tools and constructive methods we show
that if , (), then where only depend on , while if then . 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
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
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
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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