Set mappings for general graphs

TL;DR

This paper investigates set mappings in general graphs, proving bounds on the size of complete graphs needed to find subgraphs with disjoint image sets, extending classical extremal graph theory results.

Contribution

It establishes tight bounds for the existence of subgraphs with disjoint set images under a mapping, generalizing previous results for specific graph classes.

Findings

Bound N = O(m) is tight for cliques.

Bound N = O(m) is tight up to a logarithmic factor for all graphs.

Provides a systematic extension of extremal set mapping problems to general graphs.

Abstract

The study of extremal problems for set mappings has a long history. It was introduced in 1958 by Erd\H{o}s and Hajnal, who considered the case of cliques in graphs and hypergraphs. Recently, Caro, Patk\'os, Tuza and Vizer revisited this subject, and initiated the systematic study of set mapping problems for general graphs. In this paper, we prove the following result, which answers one of their questions. Let $G$ be a graph with $m$ edges and no isolated vertices and let $f : E(K_N) \rightarrow E(K_N)$ such that $f(e)$ is disjoint from $e$ for all $e \in E(K_N)$. Then for some absolute constant $C$, as long as $N \geq C m$, there is a copy $G^*$ of $G$ in $K_N$ such that $f(e)$ is disjoint from $V(G^*)$ for all $e \in E(G^*)$. The bound $N = O(m)$ is tight for cliques and is tight up to a logarithmic factor for all $G$.