Geodesic
Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^-$
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005).

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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 3-graph 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]. 
@article{DMTCS_2005_special_250_a46,
     author = {Talbot, John},
     title = {Chromatic {Tur\'an} problems and a new upper bound for the {Tur\'an} density of $\mathcal{K}_4^-$},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)},
     year = {2005},
     doi = {10.46298/dmtcs.3437},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3437/}
}
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%D 2005
%V DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05)
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Talbot, John. Chromatic Turán problems and a new upper bound for the Turán density of $\mathcal{K}_4^-$. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05), DMTCS Proceedings vol. AE, European Conference on Combinatorics, Graph Theory and Applications (EuroComb '05) (2005). doi : 10.46298/dmtcs.3437. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3437/

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