Shadows of ordered graphs

TL;DR

This paper introduces the concept of shadows in ordered graphs and proves a lower bound on shadow size, leading to strengthened results on hereditary properties of ordered graphs.

Contribution

It defines the shadow of a collection of ordered graphs and establishes a fundamental inequality, enhancing understanding of hereditary properties in ordered graphs.

Findings

Shadow of a collection of ordered graphs has size at least the collection size when the collection is small.

For hereditary properties of ordered graphs, the number of graphs of size n decreases after a certain point.

The results generalize classical isoperimetric inequalities to ordered graph settings.

Abstract

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow \d\G of a collection \G of ordered graphs, and prove the following, simple-sounding statement: if n \in \N is sufficiently large, |V(G)| = n for each G \in \G, and |\G| < n, then |\d \G| \ge |\G|. As a consequence, we substantially strengthen a result of Balogh, Bollob\'as and Morris on hereditary properties of ordered graphs: we show that if \P is such a property, and |\P_k| < k for some sufficiently large k \in \N, then |\P_n| is decreasing for k \le n < \infty.