Relative Tur\'an densities for ordered graphs: all and nothing

TL;DR

This paper advances the understanding of relative Turán densities in ordered graphs by identifying optimal host graph families and characterizing graphs with zero density, revealing nuanced differences from unordered cases.

Contribution

It introduces a family of optimal host graphs applicable to all ordered graphs and characterizes those with zero relative Turán density, enhancing theoretical understanding.

Findings

Identified a family of host graphs optimal for all ordered graphs.

Characterized ordered graphs with zero relative Turán density as those lacking a monotone path of length two.

Demonstrated the subtlety of relative Turán densities compared to unordered graphs.

Abstract

Reiher, R\"odl, Sales, and Schacht initiated the study of relative Tur\'an densities of ordered graphs and showed that it is more subtle and interesting than the unordered case. For an ordered graph $F$, its relative Tur\'an density, $\rho_{<}(F)$, is the greatest $\alpha$ such that every ordered graph $G$ has an $F$-free subgraph with at least $\alpha e(G)$ edges. This paper contains two main results about relative Tur\'an densities. First, we find a family of host graphs that is optimal for all $F$. Second, we characterise the ordered graphs with zero relative Tur\'an density: precisely those with no monotone path of length two.