On uniform eventowns

Abstract

Suppose that $n=2m$, $k=2t$ and $n>10k^7$. We show that if a family $\mathcal{F}$ of $k$-subsets of an $n$-set has only even pairwise intersections then $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{m}{t}\right)$. Moreover, every extremal family has an atomic structure. This result was previously proved by Frankl and Tokushige for $n>n_{FT}(k)$, where $n_{FT}(k)$ is at least exponential. The main technique is Delsarte linear programming.

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