Absolutely avoidable order-size pairs for induced subgraphs

TL;DR

This paper investigates pairs of integers representing subgraph sizes and edges, establishing the existence of infinitely many absolutely avoidable pairs where certain induced subgraphs cannot be avoided in large graphs.

Contribution

It proves the existence of infinitely many absolutely avoidable pairs, including a specific infinite set where the pair involves half the maximum edges, and extends results to functions close to this value.

Findings

Infinitely many absolutely avoidable pairs are shown to exist.

A specific infinite set M is identified where pairs are absolutely avoidable.

Results extend to pairs with edge counts near half the maximum, adjusted by bounded functions.

Abstract

We call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{2}$, \emph{absolutely avoidable} if there is $n_0$ such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{2}$ there is a graph on $n$ vertices and $e$ edges that contains no induced subgraph on $m$ vertices and $f$ edges. Some pairs are clearly not absolutely avoidable, for example $(m,0)$ is not absolutely avoidable since any sufficiently sparse graph on at least $m$ vertices contains independent sets on $m$ vertices. Here we show that there are infinitely many absolutely avoidable pairs. We give a specific infinite set $M$ such that for any $m\in M$, the pair $(m, \binom{m}{2}/2)$ is absolutely avoidable. In addition, among other results, we show that for any monotone integer function $q(m)$, $|q(m)|=O(m)$, there are infinitely many values of $m$ such that the pair $(m, \binom{m}{2}/2 +q(m))$ is absolutely avoidable.