Triangle-degrees in graphs and tetrahedron coverings in 3-graphs
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.