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Communication Dans Un Congrès Année : 2015

On the Zarankiewicz Problem for Intersection Hypergraphs

Résumé

Let $d$ and $t$ be fixed positive integers, and let $K^d_{t,\ldots,t}$ denote the complete $d$-partite hypergraph with $t$ vertices in each of its parts, whose hyperedges are the $d$-tuples of the vertex set with precisely one element from each part. According to a fundamental theorem of extremal hypergraph theory, due to Erd\H os~\cite{E64}, the number of hyperedges of a $d$-uniform hypergraph on $n$ vertices that does not contain $K^d_{t,\ldots,t}$ as a subhypergraph, is $n^{d-\frac{1}{t^{d-1}}}$. This bound is not far from being optimal. We address the same problem restricted to {\em intersection hypergraphs} of $(d-1)$-dimensional {\em simplices} in $\Re^d$. Given an $n$-element set $\S$ of such simplices, let $\HH^d(\S)$ denote the $d$-uniform hypergraph whose vertices are the elements of $\S$, and a $d$-tuple is a hyperedge if and only if the corresponding simplices have a point in common. We prove that if $\HH^d(\S)$ does not contain $K^d_{t,\ldots,t}$ as a subhypergraph, then its number of edges is $O(n)$ if $d=2$, and $O(n^{d-1+\epsilon})$ for any $\epsilon>0$ if $d \geq 3$. This is almost a factor of $n$ better than Erd\H os's above bound. Our result is tight, apart from the error term $\epsilon$ in the exponent. In particular, for $d=2$, we obtain a theorem of Fox and Pach~\cite{FP08}, which states that every $K_{t,t}$-free intersection graph of $n$ {\em segments} in the plane has $O(n)$ edges. The original proof was based on a separator theorem that does not generalize to higher dimensions. The new proof works in any dimension and is simpler: it uses {\em size-sensitive cuttings}, a variant of random sampling. We demonstrate the flexibility of this technique by extending the proof of the planar version of the theorem to intersection graphs of $x$-monotone curves.
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Dates et versions

hal-01188986 , version 1 (02-09-2015)

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  • HAL Id : hal-01188986 , version 1

Citer

Nabil Mustafa, Janos Pach. On the Zarankiewicz Problem for Intersection Hypergraphs. Proc. of the 23rd International Symposium on Graph Drawing and Network Visualization (GD '15), Sep 2015, Los Angeles, United States. ⟨hal-01188986⟩
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