Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs

Abstract

A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \emph{$(n,d,\hat{I}¼)$-dense} if $e_H(X,Y,Z)\ge d|X||Y||Z|-\hat{I}¼n^3$ for all $X,Y,Z\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness. In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,\hat{I}¼)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $\hat{I}{´}_1(H)$ and the minimum codegree $\hat{I}{´}_2(H)$. We actually prove a general result: define \[ f(d):=\frac{1-\sqrt{(4d-1)/3}}2. \] We prove that $(n,d,\hat{I}¼)$-density together with $\hat{I}{´}_1(H)\ge \hat{I}\pm \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)$ forces a tight Hamilton cycle whenever $d>1/3$ and $\hat{I}\pm >f(d)$. In particular, $f(1/3)=1/3$, which answers Problem~8.3(i) of Araújo, Piga and Schacht and confirms Conjecture~8.1 of Han, Shu and Wang. For the minimum codegree condition, the sharp diagonal threshold is $(\hat{I}º,\hat{I}º)$, where $\hat{I}º$ is the unique real solution of $\hat{I}º=(1-\hat{I}º{)}^3$. Since $\hat{I}º\approx 0.3177>1/4$, this gives a negative answer to Problem~8.3(ii) of Araújo, Piga and Schacht and disproves Conjecture~8.2 of Han, Shu and Wang. The two proofs use a common Hamilton-framework reduction, but the two degree conditions lead to distinct dominant-component lemmas for $(n,d,\hat{I}¼)$-dense $3$-graphs, which are of independent interest and whose proofs do not rely on the absorption method.

34 pages