An intersection theorem for systems of sets

✍ Scribed by A. V. Kostochka

• Publisher: John Wiley and Sons
• Year: 1996
• Tongue: English
• Weight: 346 KB
• Volume: 9
• Category: Article
• ISSN: 1042-9832
• DOI: 10.1002/(sici)1098-2418(199608/09)9:1/2<213::aid-rsa13>3.0.co;2-o

✦ Synopsis

Erdos and Rado defined a A-system, as a family in which every two members have the same intersection. Here we obtain a new upper bound on the maximum cardinality q ( n , q ) of an n-uniform family not containing any A-system of cardinality q. Namely, we prove that, for any a > 1 and q , there exists C = C(a, q ) such that, for any n , 0 1996 John Wiley & Sons, Inc.

INTRODUCTION

Erdos and Rado [3]

introduced the notion of a A-system. They called a family X of finite sets a A-system if every two members of X have the same intersection.

Let q ( n , q ) (respectively, q ( n , q , p ) ) denote the maximum cardinality of an n-uniform family not containing any A-system of cardinality q (respectively, a-uniform family not containing any A-system of cardinality q such that there are no p pairwise disjoint sets).

Erdos and Rado [3] proved that