Abstract

We construct a system H of exp(c log 2 n= log log n) 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 H 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). 1 Introduction Generalizing the Ray-Chaudhuri|Wilson theorem [8], Frankl and Wilson [6] proved the following intersection theorem, one of the most important results in extremal set theory: Department of Computer Science, Eotvos Un...

Citations

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