One kind of construction on sunflower with two petals*

Abstract

A sunflower (or ∆-system) with k petals and a core Y is a collection of sets S1,⋯, Sk such that Si∩Sj=Y for all i≠j; the sets S1\Y,⋯, Sk\ Y, are petals. In this paper, we first give a sufficient condition for the existence of a sunflower with 2 petals. Let F={A,B,C} be a family of subsets of a set { a1,⋯,am , b1,⋯,bn , c1,⋯,cn } with  and A={a1,⋯,am}, B={ b1,⋯,bn } and C={ c1,⋯,cn } are non-increasing lists of nonnegative integers. Suppose that for each r with    then the family F* contains a sunflower with two petals, where F*={G1 ,G2}, G1=G[Y∪X] and G2=[ Z∪X] are the subgraphs induced respectively by Y∪X and Z∪X with  for all vj Y∪X and   for all vj Z∪X. Moreover, we generalize the consequence to the case of a much more general result. Key words: Sunflower; family; tripartite graph.