Graphs with no induced \(K_{2,t}\)

Freddie Illingworth¹

¹University of Cambridge

The electronic journal of combinatorics, Tome 28 (2021) no. 1

Résumé

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 must $\alpha$ 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 Turán 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 Gyárfás, Hubenko and Solymosi and more recently Holmsen (whose argument inspired ours).

Détail

DOI : 10.37236/9223

Classification : 05C35
Mots-clés : induced Turán numbers, Ramsey number