2023/04/26 by Tokushige, Norihide
#05D05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2304.13466
Let \mathcal G be a family of subsets of an n-element set. The family \mathcal G is called 3-wise t-intersecting if the intersection of any three subsets in \mathcal G is of size at least t. For a real number p∈(0,1) we define the measure of the family by the sum of p|G|(1-p)n-|G| over all G∈\mathcal G. For example, if \mathcal G consists of all subsets containing a fixed t-element set, then it is a 3-wise t-intersecting family with the measure pt. Let 00, and let \mathcal G be a 3-wise t-intersecting family. It is known that the measure of \mathcal G is at most pt. Suppose, moreover, that \mathcal G has the measure at least (\frac12+δ)pt. We show that, by choosing t sufficiently large depending on δ, the structure of \mathcal G is one of (i) and (ii): (i) every subset in \mathcal G contains a fixed t-element set, (ii) every subset in \mathcal G contains at least t+2 elements from a fixed (t+3)-element set.