Superpolynomial Size Set-systems with Restricted Intersections mod 6 and Explicit Ramsey Graphs

Dedicated to the memory of Paul Erdős

We construct a system of subsets of a set of n elements such that the size of each set is divisible by 6 but their pairwise intersections are not divisible by 6. The result generalizes to all non-prime-power moduli m in place of m=6. This result is in sharp contrast with results of Frankl and Wilson (1981) for prime power moduli and gives strong negative answers to questions by Frankl and Wilson (1981) and Babai and Frankl (1992). We use our set-system to give an explicit Ramsey-graph construction, reproducing the logarithmic order of magnitude of the best previously known construction due to Frankl and Wilson (1981). Our construction uses certain mod m polynomials, discovered by Barrington, Beigel and Rudich (1994).