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Chromatic Turán problems and a new upper bound for the Turán density of $$\mathcal K^{-}_{4}$$. (English) Zbl 1128.05030
For positive integers $$k,r$$ and an $$r$$-graph $$\mathcal F$$, $$\text{ex}_k(n,{\mathcal F})$$ denotes the maximum number of edges in an $$\mathcal F$$-free, $$k$$-colourable $$r$$-graph on $$n$$ vertices; the Turán density of $$\mathcal F$$ is $$\pi(\mathcal F) =\lim_{n\to\infty}\frac{\text{ex}(n,{\mathcal F})}{\binom{n}{r}}$$; the $$k$$-chromatic Turán density is $$\pi_k(\mathcal F) =\lim_{n\to\infty}\frac{\text{ex}_k(n,{\mathcal F})}{\binom{n}{r}}$$. It is observed (Corollary 3) that $$\pi_k(G)$$ is known for 2-graphs from the Erdős-Simonovits-Stone theorem [P. Erdős and M. Simonovits, Stud. Sci. Math. Hung. 1, 51–57 (1966; Zbl 0178.27301)]. The 3-graph $${\mathcal K}_4^-$$ has 4 vertices and 3 edges; the problem of determining $$\pi({\mathcal K}_4^-)$$ is a special case of a question proposed in [W. G. Brown, P. Erdős and V. T. Sós, Some extremal problems on $$r$$-graphs. New Direct. Theory Graphs, Proc. Third Ann Arbor Conf., Univ. Michigan 1971, 53–63 (1973; Zbl 0258.05132)].
Theorem 4: There exists $$\omega_2>0$$ such that $$0.25682<\pi_2({\mathcal K}_4^-)<\frac3{10}-\omega_2$$.
Theorem 5: There exists $$\omega_3>0$$ such that $$\frac{5}{18}\leq\pi_3({\mathcal K}_4^-)<\frac{3+\sqrt{\frac{11}3}}{15}-\omega_3$$.
A lower bound for $$\pi({\mathcal K}_4^-)$$ was determined in [P. Frankl and Z. Füredi, Discrete Math. 50, 323–328 (1984; Zbl 0538.05050)]. Improving on upper bounds in [D. de Caen, Ars Comb. 16, 5-10 (1983; Zbl 0532.05037), and D. Mubayi, Electron. J. Comb. 10, No. 1, Research paper N10, 4 p. (2003); printed version J. Comb. 10, No. 3 (2003; Zbl 1023.05105)], the author proves Theorem 1: $$\frac27<\pi({\mathcal K}_4^-)<0.32975<\frac13-\frac1{280}$$. It is conjectured (Conjecture 1) that $$\pi_3({\mathcal K}_4^-)=\frac{5}{18}$$.

##### MSC:
 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs
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##### References:
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