Alon–Babai–Suzuki’s inequalities, Frankl–Wilson type theorem and multilinear polynomials

Let K={k1,k2,…,kr}K={k1,k2,…,kr} and L={l1,l2,…,ls}L={l1,l2,…,ls} be subsets of {0,1,…,p−1}{0,1,…,p−1} such that K∩L=0̸K∩L=0̸, where pp is a prime. Let F={F1,F2,…,Fm}F={F1,F2,…,Fm} be a family of subsets of [n]={1,2,…,n}[n]={1,2,…,n} with |Fi||Fi| (modp) ∈K∈K for all Fi∈FFi∈F and |Fi∩Fj||Fi∩Fj| (modp) ∈L∈L for any i≠ji≠j. Every subset FiFi of [n][n] can be represented by a binary code a=(a1,a2,…,an) such that aj=1aj=1 if j∈Fij∈Fi and aj=0aj=0 if j∉Fij∉Fi. Alon–Babai–Suzuki proved in non-modular version that if ki≥s−r+1ki≥s−r+1 for all ii, then |F|≤∑i=s−r+1s(ni). We generalize it in modular version. Alon–Babai–Suzuki also proved that the above bound still holds under r(s−r+1)≤p−1r(s−r+1)≤p−1 and n≥s+maxiki in modular version. Alon–Babai–Suzuki made a conjecture that if they drop one condition r(s−r+1)≤p−1r(s−r+1)≤p−1 among r(s−r+1)≤p−1r(s−r+1)≤p−1 and n≥s+maxiki, then the above bound holds. But we prove the same bound under dropping the opposite condition n≥s+maxiki. So we prove the same bound under only condition r(s−r+1)≤p−1r(s−r+1)≤p−1. This is a generalization of Frankl–Wilson theorem (Frankl and Wilson, 1981 [2]).