Erdős Problem #621

Let \(G\) be a graph on \(n\) vertices, \(\alpha_1(G)\) be the maximum number of edges that contain at most one edge from every triangle, and \(\tau_1(G)\) be the minimum number of edges that contain at least one edge from every triangle. Is it true that\[\alpha_1(G)+\tau_1(G) \leq \frac{n^2}{4}?\] A problem of Erdős, Gallai, and Tuza [EGT96], who observe that this is probably quite difficult since there are different examples where equality hold: the complete graph, the complete bipartite graph, and the graph obtained from \(K_{m,m}\) by adding one vertex joined to every other. This is true, and was proved by Norin and Sun [NoSu16], who in fact proved that\[\alpha_1(G)+\tau_B(G) \leq \frac{n^2}{4},\]where \(\tau_B(G)\) is the minimum number of edges that need to be removed to make the graph bipartite. Here \(\alpha_1(G)\) and \(\tau_1(G)\) are taken over subsets of the edge set of \(G\), and the inequality is stated multiplied through by \(4\) so that it lives in the natural numbers. The linked file states \(\tau_1\) as the least number of edges whose deletion leaves \(G\) triangle-free, which is the same as meeting every triangle of \(G\), and quantifies over an arbitrary `Fintype V` rather than `Fin n`.