Alon-babai-suzuki's conjecture related to binary codes in nonmodular version

Abstract

Let K={ k1, k2,⋯, kr} and L={ l1, l2,⋯, ls} be sets of nonnegative integers. LetF={ F1, F2,⋯, Fm} be a family of subsets of [ n ] with [ Fi]∈K for each i and | F iF∩j|∈L for any iεj. Every subset F eof [ n ] can be represented by a binary code a =(a1, a2,⋯, an) such that a i =1∈if i Fe and ai =0 if i∈ Fe. Alon et al. made a conjecture in 1991 in modular version. We prove Alon-Babai-Sukuki's Conjecture in nonmodular version. For any K and L with n s + max ki, | F |≥ (n-1s)+(n-1s-1)+⋯+(n-1s-2r+1). Copyright © 2010 K.-W. Hwang et al.