Large homogeneous submatrices

Abstract

A matrix is homogeneous if all of its entries are equal. Let P be a 2×2 zero-one matrix that is not homogeneous. We prove that if an n×n zero-one matrix A does not contain P as a submatrix, then A has an cn × cn homogeneous submatrix for a suitable constant c > 0. We further provide an almost complete characterization of the matrices P (missing only finitely many cases) such that forbidding P in A guarantees an n 1−o(1) × n 1−o(1) homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane-arrangements and string graphs.

Bipartite independence number in graphs with bounded maximum degree

“…A standard probabilistic argument shows that if H or its bipartite complement contains a cycle, then h(n, H) = O(n 1−ǫ ) for a positive ǫ. Axenovich, Tompkins, and Weber [2] proved that h(n, H) is linear in n for all but four strongly acyclic bipartite graphs H (a bipartite graph H being strongly acyclic if neither H nor its bipartite complement contains a cycle). Korándi, Pach, and Tomon [21] independently obtained similar results in the context of matrices. Note that f (n, ∆) for ∆ sublinear in n corresponds to h(n, K 1,∆+1 ).…”

Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix

“…The proof that 1.10 implies 1.14 is similar and we omit it. The result 1.8 of Korándi, Pach, and Tomon [4] has as a hypothesis that the bicomplement of G has at least τ n 2 edges. This is apparently much weaker than the hypothesis that every vertex of G has degree at most εn, but in fact the "not very dense" hypothesis is as good as the "very sparse" hypothesis, because of the next result, proved in [5].…”

“…Perhaps the situation is better for ordered bipartite graphs: certainly we are not so well-supplied with counterexamples, and there are some positive results about ordered bipartite graphs, proved recently by Korándi, Pach, and Tomon [4]. Let us say an ordered bigraph is a bigraph with linear orders on V 1 (G) and on V 2 (G).…”

Graph classes with linear Ramsey numbers

“…A number of related results appeared also in [13], where the authors study zero-one matrices that do not contain a fixed matrix as a submatrix. Primarily, they are interested in forbidden submatrices P that guarantee the existence of a square homogeneous submatrix of linear size in matrices avoiding P , where homogeneous means a submiatrix with all its entries being equal.…”

“…Primarily, they are interested in forbidden submatrices P that guarantee the existence of a square homogeneous submatrix of linear size in matrices avoiding P , where homogeneous means a submiatrix with all its entries being equal. The problems studied in [13] can be interpreted as questions about homogeneous bipartite subgraphs in colored and (vertex -)ordered bipartite graphs which do not contain a fixed forbidden colored and ordered bipartite subgraph. In this case, the notion of graph containment must preserve not only colors but also vertex order.…”