Minimum pair degree condition for tight Hamiltonian cycles in $4$-uniform hypergraphs

Abstract

We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.

Dedicated to Endre Szemerédi on the occasion of his $80^{\text{th}}$ birthday