A Generalisation of the Concentration-of-Measure Phenomenon with Applications to Intersection Problems

In this paper we prove a generalisation of the concentration-of-measure phenomenon in the discrete cube. In this setting, the concentration-of-measure phenomenon states that for every subset $\mathcal{A}$ of the discrete cube, its sum with a Hamming ball of suitably large radius $r$ -- or equivalently, its $r$-expansion -- results in a substantial increase in measure. We define a notion of `$(\gamma,C)$-well-spread' for subsets of the discrete cube $\{0,1\}^n$ for which the following holds: for all $\epsilon$, there exist constants $\gamma$ and $C$ such that for every $\mathcal{A}$ with $|\mathcal{A}| \geq \epsilon2^n$ and every $(\gamma,C)$-well-spread $S$, $|\mathcal{A} + S|$ is at least $(1-\epsilon)2^n$. We use this result to prove new non-trivial upper bounds to two intersection problems: how many subsets (or subgraphs) can one take from $[n]$ or $[\binom{n}{2}]$ such that every pair's intersection contains some given substructure? We prove non-trivial upper bounds for the $C_4$-intersection problem and the $4$-AP-intersection problem. We also give upper bounds that tend to $0$ for the $H$-intersection problem and $k$-AP-intersection problem as the number of edges and $k$ tend to infinity. Previously, non-trivial upper bounds were only known for non-bipartite $H$ and nothing was known for the $k$-AP-intersection problem.