Specified intersections

Abstract

Let M ⊂ [ n ] := { 0 , … , n } M \subset [n]:=\{0, \ldots , n\} and A {\mathcal {A}} be a family of subsets of an n n element set such that | A ∩ B | ∈ M |A \cap B| \in M for every A , B ∈ A A, B \in \mathcal {A} . Suppose that l l is the maximum number of consecutive integers contained in M M and n n is sufficiently large. Then \[ | A | > min { 1.622 n 10 2 l + 5 , 2 n / 2 + l log 2 ⁡ n } . |\mathcal {A}|> \min \{1.622^n 10^{2l+5} \, ,\quad 2^{n/2+l\log ^2 n}\}. \] The first bound complements the previous bound of roughly ( 1.99 ) n (1.99)^n due to Frankl and the second author which was proved under the assumption that M = [ n ] ∖ { n / 4 } M=[n]\setminus \{n/4\} . For l = o ( n / log 2 ⁡ n ) l=o(n/\log ^2 n) , the second bound above becomes better than the first bound. In this case, it yields 2 n / 2 + o ( n ) 2^{n/2+o(n)} , and this can be viewed as a generalization (in an asymptotic sense) of the famous Eventown theorem of Berlekamp (1969) and Graver (1975). We conjecture that our bound 2 n / 2 + o ( n ) 2^{n/2+o(n)} remains valid as long as l > n / 10 l>n/10 . Our second result complements the result of Frankl and the second author in a different direction. Fix ε > 0 \varepsilon >0 and ε n > t > n / 5 \varepsilon n > t>n/5 and let M = [ n ] ∖ ( t , t + n 0.525 ) M=[n]\setminus (t, t+n^{0.525}) . Then, in the notation above, we prove that for n n sufficiently large, \[ | A | ≤ n ( n ( n + t ) / 2 ) . |\mathcal {A}| \le n{n \choose (n+t)/2}. \] This is essentially sharp, aside from the multiplicative factor of n 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 } M=[n]\setminus \{t\} whenever ε n > t > n / 3 \varepsilon n > t > n/3 . A similar conjecture when t t is fixed and n n is large was made earlier by Frankl (1977) and proved by Frankl and Füredi (1984).