Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
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We investigate a covering problem in 3-uniform hypergraphs (3-graphs): given a 3-graph F, what is c_1(n,F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex degree δ_1(G)>d then every vertex of G is contained in a copy of F in G ? We asymptotically determine c_1(n,F) when F is the generalised triangle K_4^(3)-, and we give close to optimal bounds in the case where F is the tetrahedron K_4^(3) (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: given an n-vertex graph G with m> n^2/4 edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.
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