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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].
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
For a graph $H$ consisting of finitely many internally disjoint paths connecting two vertices, with possibly distinct lengths, we estimate the corresponding extremal number $text{ex}(n,H)$. When the lengths of all paths have the same parity, $text{ex
Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges ${a,b,c}, {c,d,e},$ and ${e,f,g}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $rle 9$. The proof of this result relies on a caref
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 wit
Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path of length $k$, is $n^{{lfloor{frac{k}{2}}rfloor}+1}$. In this paper we determine the asymptotic value o