Oleg Pikhurko

European Journal of Combinatorics

Tur\'an densities of hypergraph augmentations

For a $t$-graph $F$ and an integer $r\ge t$, the \emph{$r$-augmentation} $F(r)$ is the $r$-graph obtained by adding the same set of $r-t$ new vertices to every edge of $F$. We investigate the behaviour of the \emph{Tur\'an density $\pi(F(r))$}, which is the asymptotically maximum edge density of a large $F(r)$-free $r$...

The quadratic Brown--Erd\H{o}s--S\'os problem for 3-uniform hypergraphs with 8 and 9 edges

The famous and actively studied problem of Brown--Erd\H{o}s--S\'os from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it...

An improved bound on the minimum size of Tur\'an $(r+1,r)$-systems

For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to...