## Abstract

Let K={k_{1},k_{2},⋯,k_{r}} and L={l _{1},l_{2},⋯,l_{s}} be subsets of {0,1,⋯,p-1} such that K∩L=ø, where p is a prime. Let F={F _{1},F_{2},⋯,F_{m}} be a family of subsets of [n]={1,2,⋯,n} with |F_{i}| (modp) ∈K for all F _{i}∈F and |F_{i}∩F_{j}| (modp) ∈L for any i≈j. Every subset F_{i} of [n] can be represented by a binary code a=(a_{1},a_{2},⋯,a_{n}) such that a_{j}=1 if j∈F_{i} and a_{j}=0 if j∉F_{i}. AlonBabaiSuzuki proved in non-modular version that if k_{i}≥s-r+1 for all i, then |F|≤∑_{i=s-r+1}^{s}(ni). We generalize it in modular version. AlonBabaiSuzuki also proved that the above bound still holds under r(s-r+1)≤p-1 and n<s+max_{i}k_{i} in modular version. AlonBabaiSuzuki made a conjecture that if they drop one condition r(s-r+1)≤p-1 among r(s-r+1)≤p-1 and n<s+max_{i}k_{i}, then the above bound holds. But we prove the same bound under dropping the opposite condition n<s+max_{i}k_{i}. So we prove the same bound under only condition r(s-r+1)≤p-1. This is a generalization of FranklWilson theorem (Frankl and Wilson, 1981 [2]).

Original language | English |
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Pages (from-to) | 1477-1480 |

Number of pages | 4 |

Journal | Applied Mathematics Letters |

Volume | 24 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2011 |

## Keywords

- AlonBabaiSuzuki's inequalities
- FranklWilson theorem
- Multilinear polynomials