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Zarankiewicz numbers near the triple system threshold
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abstract
For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$. Here, we extend this by determining $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3$ and, when $m$ is large, for all $n \geq \binom{m}{2}/6+O(m)$.
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Cited by 1 Pith paper
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A congruence obstruction to Roman's bound for Zarankiewicz numbers
A congruence argument shows Roman's bound is not tight on a long interval below the design threshold, with exact values in a special case.
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