A note on induced Tur\'{a}n numbers

TL;DR

This paper asymptotically determines the maximum edges in large graphs avoiding a non-bipartite subgraph and certain induced subgraphs, expanding understanding of induced Turán numbers beyond bipartite cases.

Contribution

It extends the study of induced Turán numbers to cases where $H$ is non-bipartite and $F$ is neither an independent set nor a complete bipartite graph, providing asymptotic results.

Findings

Asymptotic determination of induced Turán numbers for non-bipartite $H$

Extension of induced Turán number theory beyond bipartite $F$

New bounds for graphs avoiding specific subgraphs

Abstract

Loh, Tait, Timmons and Zhou introduced the notion of induced Tur\'{a}n numbers, defining $\operatorname{ex}(n, \{H, F\text{-ind}\})$ to be the greatest number of edges in an $n$-vertex graph with no copy of $H$ and no induced copy of $F$. Their and subsequent work has focussed on $F$ being a complete bipartite graph. In this short note, we complement this focus by asymptotically determining the induced Tur\'{a}n number whenever $H$ is not bipartite and $F$ is not an independent set nor a complete bipartite graph.