2026/07/26 by Hao Lin, Guanghui Wang, Wenling Zhou
#math.CO
A 3-uniform hypergraph (or 3-graph) H on n vertices is (n,d,μ)-dense if eH(X,Y,Z)≥ d|X||Y||Z|-μn3 for all X,Y,Z⊆ 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,μ)-dense 3-graphs H under conditions on the minimum vertex degree δ1(H) and the minimum codegree δ2(H). We actually prove a general result: define f(d):=\frac1-√((4d-1)/3)2. We prove that (n,d,μ)-density together with δ1(H)≥α\binomn-12 forces a tight Hamilton cycle whenever d > 1/3 and α>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 (κ,κ), where κ is the unique real solution of κ=(1-κ)3. Since κ≈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, μ)-dense 3-graphs, which are of independent interest and whose proofs do not rely on the absorption method.