Abstract
The Turán 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 Turán's theorem. However, when $r\ge 3$, the problem remains open. We model the problem as an integer program and call the underlying polytope the Turán 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 Turán polytope. We also show that clique inequalities and what we call doubling inequalities are facet-defining when $r=2$. These facets lead to a simple new polyhedral proof of Turán's theorem.
Highlights
Mantel’s theorem [Man07], one of the earliest theorems in combinatorics, states that the maximum number of edges in a graph on n vertices without any triangles is n2 4, and that the maximum is attained only on complete bipartite graphs with parts of size n 2 and n 2Turan [Tur41] later generalized this theorem by showing that the maximum number of edges in a graph on n vertices without any clique of size a is at most (1 − 1 a−1 ) n2 2, and that this maximum is attained solely on complete (a − 1)-partite graphs with parts of size as equal as possible.Since many different proofs of this theorem have been found using different techniques
We prove in Theorem 6 that the maximum number of the electronic journal of combinatorics 25(3) (2018), #P3.43 edges in an a-clique free graph on n vertices is exactly n n−1 n−2 a+2 a+1 a n−2 n−3 n−4 a a−1 2
The goal is to find the maximum number of r-hyperedges in a r-uniform hypergraph on n vertices that does not contain any r-uniform hyperclique of size a
Summary
Mantel’s theorem [Man07], one of the earliest theorems in combinatorics, states that the maximum number of edges in a graph on n vertices without any triangles is n2 4. Turan [Tur41] later generalized this theorem by showing that the maximum number of edges in a graph on n vertices without any clique of size a is at most The goal is to find the maximum number of r-hyperedges in a r-uniform hypergraph (i.e., a hypergraph for which every edge is composed of r vertices) on n vertices that does not contain any r-uniform hyperclique of size a (i.e., a r-uniform hypergraph on a vertices where every set of r vertices forms an edge). This problem remains unsolved to this day. Enough, we observe that the facets we study do not get dominated as n increases, suggesting that the Turan polytope is very complex
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