Abstract
AbstractWe investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G?We asymptotically determine c1(n, F) when F is the generalized 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.
Highlights
Let F be a graph with at least one edge
What is the maximum number of edges ex(n, F) an nvertex graph can have if it does not contain a copy of F as a subgraph? This is a classical question in extremal graph theory
The extremal theory of hypergraphs has, turned out to be much harder, and even the fundamental question of determining the maximum number of edges in a 3-graph with no copy of the tetrahedron K4(3) remains open: it is the subject of a 70-year-old conjecture of Turán, and of an Erdos $1000 prize
Summary
Let F be a graph with at least one edge. What is the maximum number of edges ex(n, F) an nvertex graph can have if it does not contain a copy of F as a subgraph? This is a classical question in extremal graph theory. For every ε > 0, there exists δ > 0 and n0 ∈ N such that the following holds: for every n n0, if H is a 3-graph on n + 1 vertices with minimum vertex-degree at least (c − δ)(n2/2) and x ∈ V(H) is not covered by a copy of K4(3)− in H, the link graph Hx can be made bipartite by removing at most εn edges. Where there is no risk of confusion, we identify hypergraphs with their edge-sets
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