A characterization of edge-ordered graphs with almost linear extremal functions

TL;DR

This paper proves a near-linear upper bound for extremal functions of edge-ordered forests with chromatic number 2, advancing understanding of Turán-type problems in edge-ordered graphs.

Contribution

It resolves a conjecture by establishing a stronger upper bound of $n2^{O(\sqrt{\log n})}$ for extremal functions of certain edge-ordered forests.

Findings

Proved the conjecture with an upper bound of $n2^{O(\sqrt{\log n})}$.

Identified a gap in extremal functions for edge-ordered graphs.

Conjectured an even stronger upper bound of $n\log^{O(1)}n$.

Abstract

The systematic study of Tur\'an-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. arXiv:2001.00849. They conjectured that the extremal functions of edge-ordered forests of order chromatic number 2 are $n^{1+o(1)}$. Here we resolve this conjecture proving the stronger upper bound of $n2^{O(\sqrt{\log n})}$. This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions $\Omega(n^c)$ for some $c>1$. However, our result is probably not the last word: here we conjecture that the even stronger upper bound of $n\log^{O(1)}n$ also holds for the same set of extremal functions.