THE DENSITY TURÁN PROBLEM

Let H be a graph on n vertices and let the blow-up graph G[H] be defined as follows. We replace each vertex vi of H by a cluster Ai and connect some pairs of vertices of Ai and Aj if (vi,vj) is an edge of the graph H. As usual, we define the edge density between Ai and Aj as d(Ai,Aj)= e(Ai,Aj)/|Ai||Aj|. We study the following problem. Given densities γij for each edge (i,j) ∈E(H), one has to decide whether there exists a blow-up graph G[H], with edge densities at least γij, such that one cannot choose a vertex from each cluster, so that the obtained graph is isomorphic to H, i.e., no H appears as a transversal in G[H]. We call dcrit(H) the maximal value for which there exists a blow-up graph G[H] with edge densities d(Ai,Aj)=d crit(H) ((vi,vj) ∈E(H)) not containing H in the above sense. Our main goal is to determine the critical edge density and to characterize the extremal graphs. First, in the case of tree T we give an efficient algorithm to decide whether a given set of edge densities ensures the existence of a transversal T in the blow-up graph. Then we give general bounds on dcrit(H) in terms of the maximal degree. In connection with the extremal structure, the so-called star decomposition is proved to give the best construction for H-transversal-free blow-up graphs for several graph classes. Our approach applies algebraic graph-theoretical, combinatorial and probabilistic tools. © 2012 Copyright Cambridge University Press

The density Turán problem for hypergraphs

Given a k-graph H a complete blow-up of H is a k-graph Hˆ formed by replacing each v ∈ V (H) by a non-empty vertex class Av and then inserting all edges between any k vertex classes corresponding to an edge of H. Given a subgraph G ⊆ Hˆ and an edge e ∈ E(H) we define the density de(G) to be the proportion of edges present in G between the classes corresponding to e. The density Tur´an problem for H asks: determine the minimal value dcrit(H) such that any subgraph G ⊆ Hˆ satisfying de(G) > dcrit(H) for every e ∈ E(H) contains a copy of H as a transversal, i.e. a copy of H meeting each vertex class of Hˆ exactly once. We give upper bounds for this hypergraph density Tur´an problem that generalise the known bounds for the case of graphs due to Csikv´ari and Nagy [3], although our methods are different, employing an entropy compression argument