Specified Intersections

Let M ⊂ [n] := {0, . . . , n} and A be a family of subsets of an n element set such that |A∩B| ∈M for every A,B ∈ A. Suppose that l is the maximum number of consecutive integers contained in M and n is sufficiently large. Then |A| < min{1.62210 , 2 log 2 }. The first bound complements the previous bound of roughly (1.99)n due to Frankl and the second author [9] proved under the assumption that M = [n] \{n/4}. For l = o(n/ log n), the second bound above becomes better than the first bound. In this case, it yields 2n/2+o(n) and this can be viewed as a generalization (in an asymptotic sense) of the famous Eventown theorem of Berlekamp [2] and Graver [12]. We conjecture that our bound 2n/2+o(n) remains valid as long as l < n/10. Our second result complements the result of [9] in a different direction. Fix ε > 0 and εn < t < n/5 and let M = [n] \ (t, t + n0.525). Then, in the notation above, we prove that for n sufficiently large, |A| ≤ n ( n (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. We conjecture that a similar bound holds for M = [n] \ {t} whenever εn < t < n/3. A similar conjecture when t is fixed and n is large was earlier made by Frankl [6] and proved by Frankl and Füredi [7]. ∗Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL 60607. Research supported in part by NSF grant DMS-0969092. Email: [email protected] †Department of Mathematics and Computer Science, Emory University. Research supported in part by NSF grant DMS-0800070. Email: [email protected]