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 $(p,μ)$-dense if $e_H(X,Y,Z)\ge p|X||Y||Z|-μn^3$ for all $X,Y,Z\subseteq V(H)$. Ara{ú}jo, Piga and Schacht asked whether $p,α>1/4$ together with $δ_2(H)\geαn$ force a tight Hamilton cycle. We give a negative answer: for every $\varepsilon,μ>0$ and all sufficiently large $n$, there exists an $n$-vertex $(p_0-\varepsilon,μ)$-dense 3-graph $H$ with $δ_2(H)\ge(p_0-\varepsilon)n$ and no tight Hamilton cycle, where $p_0:=\max_{0\le x\le1}\min\{x^3,1-x\}\approx0.317672$. For every $p>1/3$, we determine the asymptotically sharp minimum-codegree threshold. Writing \[ δ_0(p)= \left(\frac{1-\sqrt{(4p-1)/3}}{2}\right)^2, \] we prove that every sufficiently large $(p,μ)$-dense 3-graph $H$ with $δ_2(H)\geαn$ contains a tight Hamilton cycle whenever $α>δ_0(p)$ and $μ$ is sufficiently small. A matching construction shows that this threshold is best possible. The proof uses absorption together with a new fixed-length connecting lemma based on a regular slice, a directed pair-state graph, and a finite scalar lemma.