2026/07/23 by Xichao Shu · 1 citation
#math.CO
We study tight Hamilton cycles in linearly quasirandom 3-graphs. An n-vertex 3-graph H is (p,μ)-dense if eH(X,Y,Z)≥ p|X||Y||Z|-μn3 for all X,Y,Z⊆ V(H). Araújo, Piga and Schacht asked whether p,α>1/4 together with δ2(H)≥αn force a tight Hamilton cycle. We give a negative answer: for every ε,μ>0 and all sufficiently large n, there exists an n-vertex (p0-ε,μ)-dense 3-graph H with δ2(H)≥(p0-ε)n and no tight Hamilton cycle, where p0:=max0≤ x≤1min\x3,1-x\≈0.317672. For every p>1/3, we determine the asymptotically sharp minimum-codegree threshold. Writing δ0(p)= ((1-√((4p-1)/3))/(2))2, we prove that every sufficiently large (p,μ)-dense 3-graph H with δ2(H)≥αn contains a tight Hamilton cycle whenever α>δ0(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.