Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs

We study tight Hamilton cycles in linearly quasirandom $3$-graphs. An $n$-vertex $3$-graph $H$ is $(p,\mu )$-dense if $e_H(X,Y,Z)\ge p\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }\mathrm{\mid }Z\mathrm{\mid }-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. Ara'ujo, Piga and Schacht asked whether the conditions $p,\alpha >1\mathrm{/}4$ and ${\delta }_2(H)\ge \alpha n$ force a tight Hamilton cycle. We give a negative answer to this question. More generally, we determine the asymptotically sharp minimum-codegree threshold for the existence of a tight Hamilton cycle for every density $p\in (0,1)$. The resulting threshold is a discontinuous piecewise-defined function with four distinct regimes, and matching constructions show that every piece is best possible. The proof combines the absorption method and a fixed-length connecting lemma above density $1\mathrm{/}3$ with a canonical-component Hamilton framework at and below density $1\mathrm{/}3$.