## Abstract

Let K={k_{1},k_{2},…,k_{r}} and L={l_{1},l_{2},…,l_{s}} be disjoint subsets of {0,1,…,p−1}, where p is a prime and A={A_{1},A_{2},…,A_{m}} is a family of subsets of [n]={1,2,…,n} such that |A_{i}|(modp)∈K for all A_{i}∈A and |A_{i}∩A_{j}|(modp)∈L for i≠j. In 1991, Alon, Babai and Suzuki conjectured that if n≥s+max_{1≤i≤r}k_{i}, then |A|≤[Formula presented]+[Formula presented]+⋯+[Formula presented]. In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition n≥2s−r. In 2015, Hwang and Kim verified this conjecture. In this paper, we will prove that if n≥2s−2r+1 or n≥s+max_{1≤i≤r}k_{i}, then |A|≤[Formula presented]+[Formula presented]+⋯+[Formula presented].This result strengthens both the upper bound of Alon–Babai–Suzuki's conjecture and Qian and Ray-Chaudhuri's result, when n≥2s−2.

Original language | English |
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Pages (from-to) | 109-118 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 341 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2018 |

Externally published | Yes |

## Keywords

- Alon–Babai–Suzuki's conjecture
- Extremal set theory
- Frankl–Ray-Chaudhuri–Wilson theorems
- Restricted intersections

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics