Graphs with no induced $K_{2,t}

Electron. J. Comb. 28 (2021), P1.19 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 fo...

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Bibliographic Details
Main Author Illingworth, Freddie
Format Journal Article
LanguageEnglish
Published 17.12.2019
Subjects
Mathematics - Combinatorics
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Summary:Electron. J. Comb. 28 (2021), P1.19 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).
DOI:10.48550/arxiv.1912.07970