# On set systems with restricted intersections modulo a composite number

## Abstract

Let *S* be a set of *n* elements, and let *H* be a set-system on *S*, which satisfies that the size of any element of *H* is divisible by *m* but the intersection of any two elements of *H* is not divisible by *m* If *m* is a prime or prime-power, then the famous Frankl-Wilson theorem [3] implies that |*H*| = *O*(*n*^{m-1}), i.e. for fixed *m*, its size is at most polynomial in *n*. This theorem has numerous applications in combinatorics and also in geometry, (c.f. the disproof of *Borsuk's conjecture* by *Kahn* and *Kalai* in 1992 ([4]), or explicit constructions of Ramsey graphs, or other geometric applications related to the Hadwiger-problem.) *Frankl* and *Wilson* asked in [3] whether an analogous upper bound existed for *non-prime power, composite moduli*. Here we show a surprising construction of a *superpolynomial-sized* uniform set-system *H* satisfying the intersection-property, for every non-prime-power, composite *m*, negatively settling a related conjecture of *Babai* and *Frankl* [1]. The proof uses a low-degree polynomial-construction of *Barrington, Beigel* and *Rudich* [2], and a new method (Lemma 8), for constructing set-systems from multivariate polynomials.

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## References

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