  We study sufficient conditions for the existence of Hamilton cycles in
uniformly dense $3$-uniform hypergraphs. Problems of this type were first
considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and
Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly
strong notion of uniformly dense hypergraphs. We focus on tight cycles and
obtain optimal results for a weaker notion of uniformly dense hypergraphs.
  We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the
property that for any set of vertices $X$ and for any collection $P$ of pairs
of vertices, the number of hyperedges composed by a pair belonging to $P$ and
one vertex from $X$ is at least $(1/4+o(1))|X||P| - o(|V|^3)$ and $H$ has
minimum vertex degree at least $\Omega(|V|^2)$, then $H$ contains a tight
Hamilton cycle. A probabilistic construction shows that the constant $1/4$ is
optimal in this context.
