On the Turán number of ordered forests

An ordered graph H is a graph with a linear ordering on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number exn1+ε for some positive ε=ε(H) unless H is a forest that has a bipartition V1∪V2 such that V1 totally precedes V2 in the ordering. Making progress towards a conjecture of Pach and Tardos, we prove that ex<(n,H)=n1+o(1) holds for all such forests that are “degenerate” in a certain sense. This class includes every forest for which an n1+o(1) upper bound was previously known, as well as new examples. For example, the class contains all forests with |V1|≤3. Our proof is based on a density-increment argument. © 2017 Elsevier B.V

On the Turán number of ordered forests

An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number exn1+ε for some positive ε=ε(H) unless H is a forest that has a proper 2-coloring with one color class totally preceding the other one. Making progress towards a conjecture of Pach and Tardos, we prove that ex<(n,H)=n1+o(1) holds for all such forests that are “degenerate” in a certain sense. This class includes every forest for which an n1+o(1) upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument. © 2019 Elsevier Inc

Large homogeneous submatrices

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 a 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 n1 - o(1) × n1 - o(1) homogeneous submatrix. We apply our results to chordal bipartite graphs, totally balanced matrices, halfplane arrangements, and string graphs. © 2020 author