**Vince Grolmusz
**

*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).

- Introduction
- Preliminaries
- The Lower Bound
- An Application: Ramsey Graphs
- Open Problems
- Bibliography
- About this document ...