John Talbot

Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^$
dmtcs:3437 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2005,
DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)

https://doi.org/10.46298/dmtcs.3437
Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^$
Authors: John Talbot ^{1}
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John Talbot
1 Department of Mathematics
We consider a new type of extremal hypergraph problem: given an $r$graph $\mathcal{F}$ and an integer $k≥2$ determine the maximum number of edges in an $\mathcal{F}$free, $k$colourable $r$graph on $n$ vertices. Our motivation for studying such problems is that it allows us to give a new upper bound for an old problem due to Turán. We show that a 3graph in which any four vertices span at most two edges has density less than $\frac{33}{ 100}$, improving previous bounds of $\frac{1}{ 3}$ due to de Caen [1], and $\frac{1}{ 3}4.5305×10^6$ due to Mubayi [9].