A hypergraph extension of the bipartite Turán problem

Hypergraph Turán Problems

One of the earliest results in Combinatorics is Mantel’s theorem from 1907 that the largest triangle-free graph on a given vertex set is complete bipartite. However, a seemingly similar question posed by Turán in 1941 is still open: what is the largest 3-uniform hypergraph on a given vertex set with no tetrahedron? This question can be considered a test case for the general hypergraph Turán pro...

The density Turán problem for some 3-uniform unihypercyclic linear hypergraphs An efficient testing algorithm

Let H = (V, E) be a 3-uniform linear hypergraph with one hypercycle C3. We consider a blow-up hypergraph B[H]. We are interested in the following problem. We have to decide whether there exists a blow-up hypergraph B[H] of the hypergraph H, with hyperedge densities satisfying special conditions, such that the hypergraph H appears in a blow-up hypergraph as a transversal. We present an efficient...

Turán Problems and Shadows III: Expansions of Graphs

The expansion G of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let ex3(n, F ) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F . The study of ex3(n,G ) includes some well-researched...

An Ordered Turán Problem for Bipartite Graphs

Let F be a graph. A graph G is F -free if it does not contain F as a subgraph. The Turán number of F , written ex(n, F ), is the maximum number of edges in an F -free graph with n vertices. The determination of Turán numbers of bipartite graphs is a challenging and widely investigated problem. In this paper we introduce an ordered version of the Turán problem for bipartite graphs. Let G be a gr...

On a Turán-type hypergraph problem of Brown, Erdős and T. Sós