Cycle decompositions in 3-uniform hypergraphs (Q6057482)

Let \(H=(V,E)\) be a \(k\)-uniform hypergraph. For \(v\in V\), \(\text{deg}_H(v)\) is the number of edges containing \(v\); for a positive integer \(k\), \(H\) is \(k\)-vertex-divisible when \(k\vert \text{deg}_H(v)\) for every \(v\in V\). The codegree, \(\text{deg}_H(S)\), of a set \(S\subseteq V\) with \(\vert S\vert =k-1\) is the number of edges of \(H\) that contain all vertices in \(S\); the minimum and maximum codegrees of \(H\) over all \(S\) are \(\delta_{k-1}(H)\) and \(\Delta_{k-1}(H)\). A decomposition of \(H=(V,E)\) is a collection of subgraphs of \(H\) whose edge-sets partition \(E\); where all subgraphs in the decomposition are isomorphic to a hypergraph \(F\), it is an \(F\)-decomposition and \(H\) is \(F\)-decomposable. For \(k\ge2\) and \(\ell\ge k+1\), the \(k\)-uniform tight cycle of length \(\ell\) is the \(k\)-graph \(C^k_\ell\) whose vertices are \(\{v_1,v_2,\dots,v_\ell\}\) and whose edges are all \(k\)-sets of consecutive vertices of the form \(\{v_i, v_{i+1},\dots,v_{i+k-1}\}\) for \(1\le i\le\ell\) (indices modulo \(\ell\)); the authors define \(C^k_\ell\)-divisiblity. The \(C^k_\ell\)-decomposition threshold \(\delta_{C^k_\ell}(n)\) is the minimum \(d\) such that every \(C^k_\ell\)-divisible \(k\)-graph \(H\) on \(n\) vertices with \(\delta_{k-1}(H)\ge d\) is \(C^k_\ell\)-decomposable; \(\delta_{C^k_\ell}\) denotes \(\limsup\limits_{n\to\infty}\frac{\delta_{C^k_\ell}(n)}{n}\). The ``main result'' is Theorem 1.1: Suppose \(\ell\) satisfies one of the following: (i) \(\ell\) is divisible by 3 and at least 9, or (ii) \(\ell\ge107\). Then \(\delta_{C^3_\ell}=\frac23\). \S9 contains open questions.