Tur\'an problems for Edge-ordered graphs
Abstract
In this paper we initiate a systematic study of the Tur\'an problem for edge-ordered graphs. A simple graph is called edge-ordered, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph G avoids another edge-ordered graph H, if no subgraph of G is isomorphic to H. The Tur\'an number of an edge-ordered graph H is the maximum number of edges in an edge-ordered graph on n vertices that avoids H. We study this problem in general, and establish an Erdos-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Tur\'an number of an edge-ordered graph is its order chromatic number. We establish several important properties of this parameter. We also study Tur\'an numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry.