Set mappings for general graphs

TL;DR

This work resolves a central question on set-mapping problems for general graphs by proving that every graph $G$ with $m$ edges and no isolated vertices embeds into a large complete graph $K_N$, avoiding all forbidden $f$-images, whenever $N \ge C m$ for an absolute constant $C$. The proof combines a probabilistic construction with a deterministic embedding procedure: it selects structured candidate sets and uses a greedy embedding that avoids all disallowed images, controlled via Chernoff-type bounds and union bounds. Consequently, $w(G) = O(m)$ for all graphs $G$, with tightness known up to a logarithmic factor, and the $K_n$ case recovers the known $\Theta(n^2)$ scaling; the paper also sketches hypergraph extensions and several open problems. These results advance understanding of how set mappings constrain subgraph appearances and introduce tools that may apply to related extremal problems in combinatorics.

Abstract

The study of extremal problems for set mappings has a long history. It was introduced in 1958 by Erdős and Hajnal, who considered the case of cliques in graphs and hypergraphs. Recently, Caro, Patkós, 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 $G$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Conjecture 3.2
• Theorem 3.3: Lovász local lemma, see Chapter 5 in AlonSpencer
• Proposition 3.4
• proof