The maximum size of 3-uniform hypergraphs not containing a fano plane
Journal of Combinatorial Theory Series B
On the Turán number of triple systems
Journal of Combinatorial Theory Series A
Extremal Graph Theory
Stability theorems for cancellative hypergraphs
Journal of Combinatorial Theory Series B
Triple Systems Not Containing a Fano Configuration
Combinatorics, Probability and Computing
The Turán Number Of The Fano Plane
Combinatorica
On A Hypergraph Turán Problem Of Frankl
Combinatorica
The Turán problem for projective geometries
Journal of Combinatorial Theory Series A
A hypergraph extension of Turán's theorem
Journal of Combinatorial Theory Series B
Journal of Combinatorial Theory Series A
Note: Quadruple systems with independent neighborhoods
Journal of Combinatorial Theory Series A
Combinatorial Problems for Horn Clauses
Graph Theory, Computational Intelligence and Thought
Almost all triple systems with independent neighborhoods are semi-bipartite
Journal of Combinatorial Theory Series A
Counting substructures II: Hypergraphs
Combinatorica
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Let ${\cal H}$ be a 3-uniform hypergraph on an $n$-element vertex set $V$. The neighbourhood of $a,b\in V$ is $N(ab):= \{x: abx\in E({\cal H})\} $. Such a 3-graph has independent neighbourhoods if no $N(ab)$ contains an edge of ${\cal H}$. This is equivalent to ${\cal H}$ not containing a copy of $\mathbb{F} :=\{ abx$, $aby$, $abz$, $xyz\}$.In this paper we prove an analogue of the Andrásfai–Erdös–Sós theorem for triangle-free graphs with minimum degree exceeding $2n/5$. It is shown that any $\mathbb{F}$-free 3-graph with minimum degree exceeding $(\frac{4}{9}-\frac{1}{125})\binom{n}{2}$ is bipartite, (for $n n_0$), i.e., the vertices of ${\cal H}$ can be split into two parts so that every triple meets both parts.This is, in fact, a Turán-type result. It solves a problem of Erdös and T.Sós, and answers a question of Mubayi and Rödl that\[\ex(n,\mathbb{F}_{3,2})=\max\limits_{\alpha}(n-\alpha)\left({\alpha\atop2}\right)\].Here the right-hand side is $\frac{4}{9}\binom{n}{3}+O(n^2)$. Moreover $e({\cal H})={\rm ex}(n,\mathbb{F})$ is possible only if $V({\cal H})$ can be partitioned into two sets $A$ and $B$ so that each triple of ${\cal H}$ intersects $A$ in exactly two vertices and $B$ in one.