The degree and codegree threshold for linear triangle covering in 3-graphs

Given two k-uniform hypergraphs F and G, we say that G has an F-covering if every vertex in G is contained in a copy of F. For 1 <= i <= k - 1, let c(i)(n, F) be the least integer such that every n-vertex k-uniform hypergraph G with delta(i)(G) > c(i)(n, F) has an F-covering. The covering problem has been systematically studied by Falgas-Ravry and Zhao [Codegree thresholds for covering 3-uniform hypergraphs, SIAM J. Discrete Math., 2016]. In 2021, Falgas-Ravry, Markstrom, and Zhao [Triangle degrees in graphs and tetrahedron coverings in 3-graphs, Combinatorics, Probability and Computing, 2021] asymptotically determined c(1)(n, F) when F is the generalized triangle. In this note, we give the exact value of c(2)(n, F) and asymptotically determine c(1)(n, F) when F is the linear triangle C-6(3), where C-6(3) is the 3-uniform hypergraph with vertex set {v(1), v(2), v(3), v(4), v(5), v(6)} and edge set {v(1)v(2)v(3), v(3)v(4)v(5), v(5)v(6)v(1)}.