Graphs with no induced $K_{2,t}$

TL;DR

This paper investigates the structure of graphs with a given density that avoid induced $K_{2,t}$ subgraphs, providing new bounds and generalizations related to induced Turán numbers and large clique existence.

Contribution

It offers new results on the minimum edge density needed to guarantee large cliques or subgraphs in graphs avoiding induced $K_{2,t}$, extending previous work and connecting to induced Turán numbers.

Findings

Results for graphs with edge density bounded away from zero.

Results for graphs with edge density tending to zero, related to induced Turán numbers.

Generalization of previous theorems on induced subgraph containment.

Abstract

Consider a graph $G$ on $n$ vertices with $\alpha \binom{n}{2}$ edges which does not contain an induced $K_{2, t}$ ($t \geqslant 2$). How large does $\alpha$ have to be to ensure that $G$ contains, say, a large clique or some fixed subgraph $H$? We give results for two regimes: for $\alpha$ bounded away from zero and for $\alpha = o(1)$. Our results for $\alpha = o(1)$ are strongly related to the Induced Tur\'{a}n numbers which were recently introduced by Loh, Tait, Timmons and Zhou. For $\alpha$ bounded away from zero, our results can be seen as a generalisation of a result of Gy\'{a}rf\'{a}s, Hubenko and Solymosi and more recently Holmsen (whose argument inspired ours).