### Chromatic Turán problems and a new upper bound for the Turán density of K4-

*John Talbot*

#### Abstract

We consider a new type of extremal hypergraph problem: given an r-graph F and an integer k≥2 determine the maximum number of edges in an 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 3-graph in which any four vertices span at most two edges has density less than 33 / 100, improving previous bounds of 1 / 3 due to de Caen [deC], and 1 / 3-4.5305×10-6 due to Mubayi [M].

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