A note on codegree problems for hypergraphs (Q2732632)

The codegree, \(\text{codeg}(\{u,v\})\), of distinct vertices \(u\), \(v\) of a triple system (= 3-uniform hypergraph) \(\mathcal G\) is defined to be \(|\{e\in{\mathcal G}\mid u, v\in e\}|\); codeg\(({\mathcal G})\) is defined to be \(\min_{u\neq v}\) \({\text{codeg}}(\{u,v\})\). The authors consider a question of S. Abbasi, Problem 1.2: When \(n\equiv 0\pmod 4\), does every triple system \(\mathcal G\) on \(n\) vertices satisfying \({\text{ codeg}}({\mathcal G})\geq\frac n2\), admit a covering by vertex-disjoint copies of \(K_4^{(3)}\)? They prove Theorem 2.1: For every \(\varepsilon>0\) there exists an infinite family \(\{{\mathcal G}_i\}_{i\in I}\) of triple systems so that each \({\mathcal G}\in\{{\mathcal G}_i\}_{i\in I}\) has the property that \(\text{codeg} ({\mathcal G})\geq (\frac 35-\varepsilon)|V({\mathcal G})|\), and that every collection of vertex-disjoint copies of \(K_4^{(3)}\) within \(\mathcal G\) leaves at least \(\varepsilon|V({\mathcal G})|\) vertices uncovered. They follow this with Conjecture 3.1: Let \(\mathcal G\) be a triple system on \(n\) vertices. If \({\text{codeg}}({\mathcal G})\geq\frac{n}2\), then \(\mathcal G\) contains a \(K_4^{(3)}\). Modifying an example due to the second author and \textit{V. Rödl} [Regularity properties for triple systems, in preparation], they prove probabilistically that the constant \(\frac 12\) in the lower bound for \({\text{codeg}}({\mathcal G})\) in Conjecture 3.1 cannot be improved: There exists a triple system \({\mathcal G}\) on \(n\) vertices which satisfies \(\text{codeg}({\mathcal G})=\frac n2-o(n)\), but which contains no \(K_4^{(3)}\).