The Tur\'an Polytope

TL;DR

This paper introduces the Turán polytope to model the hypergraph Turán problem, revealing new facet-defining inequalities and providing a polyhedral proof of Turán's theorem, especially for the case when r=2.

Contribution

It models the Turán problem as an integer program, identifies new facet-defining inequalities, and offers a polyhedral proof of Turán's theorem.

Findings

Generalized web and wheel inequalities are facet-defining for the Turán polytope.

Clique and doubling inequalities are facet-defining for r=2.

A new polyhedral proof of Turán's theorem is provided.

Abstract

The Tur\'an hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in Tur\'an's theorem. However, when $r\geq 3$, the problem remains open. We model the problem as an integer program and call the underlying polytope the Tur\'an polytope. We draw parallels between the latter and the stable set polytope: we show that generalized and transformed versions of the web and wheel inequalities are also facet-defining for the Tur\'an polytope. We also show clique inequalities and what we call doubling inequalities are facet-defining when $r=2$. These facets lead to a simple new polyhedral proof of Tur\'an's theorem.