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Incompatible intersection properties (1808.01229v1)
Published 3 Aug 2018 in math.CO and cs.DM
Abstract: Let $\mathcal F\subset 2{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $|\mathcal F|\le 2{n-2}$ is proved. We believe that $38$ can be replaced by $3$ and provide a simple-looking conjecture that would imply this.