A strengthened inequality of Alon–Babai–Suzuki's conjecture on set systems with restricted intersections modulo p

Xin Wang; Hengjia Wei; Gennian Ge

doi:10.1016/j.disc.2017.08.019

A strengthened inequality of Alon–Babai–Suzuki's conjecture on set systems with restricted intersections modulo p

Xin Wang, Hengjia Wei, Gennian Ge

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Abstract

Let K={k₁,k₂,…,k_r} and L={l₁,l₂,…,l_s} be disjoint subsets of {0,1,…,p−1}, where p is a prime and A={A₁,A₂,…,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≤rk_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≤rk_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
Pages (from-to)	109-118
Number of pages	10
Journal	Discrete Mathematics
Volume	341
Issue number	1
DOIs	https://doi.org/10.1016/j.disc.2017.08.019
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

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10.1016/j.disc.2017.08.019

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title = "A strengthened inequality of Alon–Babai–Suzuki's conjecture on set systems with restricted intersections modulo p",

abstract = "Let K={k1,k2,…,kr} and L={l1,l2,…,ls} be disjoint subsets of {0,1,…,p−1}, where p is a prime and A={A1,A2,…,Am} is a family of subsets of [n]={1,2,…,n} such that |Ai|(modp)∈K for all Ai∈A and |Ai∩Aj|(modp)∈L for i≠j. In 1991, Alon, Babai and Suzuki conjectured that if n≥s+max1≤i≤rki, 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+max1≤i≤rki, 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.",

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

author = "Xin Wang and Hengjia Wei and Gennian Ge",

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T1 - A strengthened inequality of Alon–Babai–Suzuki's conjecture on set systems with restricted intersections modulo p

AU - Wang, Xin

AU - Wei, Hengjia

AU - Ge, Gennian

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N2 - Let K={k1,k2,…,kr} and L={l1,l2,…,ls} be disjoint subsets of {0,1,…,p−1}, where p is a prime and A={A1,A2,…,Am} is a family of subsets of [n]={1,2,…,n} such that |Ai|(modp)∈K for all Ai∈A and |Ai∩Aj|(modp)∈L for i≠j. In 1991, Alon, Babai and Suzuki conjectured that if n≥s+max1≤i≤rki, 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+max1≤i≤rki, 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.

AB - Let K={k1,k2,…,kr} and L={l1,l2,…,ls} be disjoint subsets of {0,1,…,p−1}, where p is a prime and A={A1,A2,…,Am} is a family of subsets of [n]={1,2,…,n} such that |Ai|(modp)∈K for all Ai∈A and |Ai∩Aj|(modp)∈L for i≠j. In 1991, Alon, Babai and Suzuki conjectured that if n≥s+max1≤i≤rki, 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+max1≤i≤rki, 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.

KW - Alon–Babai–Suzuki's conjecture

KW - Extremal set theory

KW - Frankl–Ray-Chaudhuri–Wilson theorems

KW - Restricted intersections

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A strengthened inequality of Alon–Babai–Suzuki's conjecture on set systems with restricted intersections modulo p

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