Specified Intersections

TL;DR

This paper establishes new upper bounds on the size of set families with intersection sizes in a specified subset, extending classical theorems and providing asymptotically sharp results using combinatorial and number-theoretic techniques.

Contribution

It introduces novel bounds on intersection families, generalizing the Eventown theorem and complementing prior results with asymptotically optimal estimates.

Findings

New upper bounds for intersection families depending on the structure of M.

Extension of the Eventown theorem to more general intersection conditions.

Bounds are sharp up to multiplicative factors, using combinatorial and number theory methods.

Abstract

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is sufficiently large. Then we prove that |F| < min {1.622^n 100^l, 2^{n/2+l log^2 n}}. The first bound complements the previous bound of roughly (1.99)^n due to Frankl and the second author which applies even when M={0, 1,.., n} - {n/4}. For small l, the second bound above becomes better than the first bound. In this case, it yields 2^{n/2+o(n)} and this can be viewed as a generalization (in an asymptotic sense) of the famous Eventown theorem of Berlekamp. Our second result complements the result of Frankl-Rodl in a different direction. Fix eps>0 and eps n < t < n/5 and let M={0, 1, .., n)-(t, t+n^{0.525}). Then, in the notation above, we prove that for n sufficiently large, |F| < n{n \choose (n+t)/2}. This is essentially sharp aside from the multiplicative factor of n. The short proof uses the Frankl-Wilson theorem and results about the distribution of prime numbers.