The profile polytope of non-trivial intersecting families

TL;DR

This paper characterizes the extreme points of profile vectors for non-trivial intersecting families of subsets, providing insight into their structural properties in combinatorics.

Contribution

It determines the extreme points of the profile vector set for non-trivial intersecting families, a novel structural result in combinatorics.

Findings

Identified the extreme points of profile vectors for non-trivial intersecting families.

Provided a characterization of the structure of these families.

Enhanced understanding of the combinatorial properties of intersecting families.

Abstract

The profile vector of a family $\mathcal{F}$ of subsets of an $n$-element set is $(f_0,f_1, \ldots, f_n)$ where $f_i$ denotes the number of the $i$-element members of $\mathcal{F}$. In this paper we determine the extreme points of the set of profile vectors for the class of non-trivial intersecting families.