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Register a new userThe maximal length of a gap between r-graph Turan densities
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Oleg Pikhurko
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The Turan density $pi(cal F)$ of a family $cal F$ of $r$-graphs is the limit as $ntoinfty$ of the maximum edge density of an $cal F$-free $r$-graph on $n$ vertices. Erdos [Israel J. Math 2 (1964) 183--190] proved that no Turan density can lie in the open interval $(0,r!/r^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Turan densities has strictly smaller length. In particular, this implies a conjecture of Grosu [E-print arXiv:1403.4653v1, 2014].
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Let $rge 3$. Given an $r$-graph $H$, the minimum codegree $delta_{r-1}(H)$ is the largest integer $t$ such that every $(r-1)$-subset of $V(H)$ is contained in at least $t$ edges of $H$. Given an $r$-graph $F$, the codegree Turan density $gamma(F)$ is the smallest $gamma >0$ such that every $r$-graph on $n$ vertices with $delta_{r-1}(H)ge (gamma + o(1))n$ contains $F$ as a subhypergraph. Using results on the independence number of hypergraphs, we show that there are constants $c_1, c_2>0$ depending only on $r$ such that [ 1 - c_2 frac{ln t}{t^{r-1}} le gamma(K_t^r) le 1 - c_1 frac{ln t}{t^{r-1}}, ] where $K_t^r$ is the complete $r$-graph on $t$ vertices. This gives the best general bounds for $gamma(K_t^r)$.
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Let $f(n,H)$ denote the maximum number of copies of $H$ possible in an $n$-vertex planar graph. The function $f(n,H)$ has been determined when $H$ is a cycle of length $3$ or $4$ by Hakimi and Schmeichel and when $H$ is a complete bipartite graph with smaller part of size 1 or 2 by Alon and Caro. We determine $f(n,H)$ exactly in the case when $H$ is a path of length 3.
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