Graphs with α₁ and τ₁ both large

Given a graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$. Erd\H{o}s, Gallai, and Tuza introduced several problems with the unifying theme that $\alpha_1(G)$ and $\tau_1(G)$ cannot both be "very large"; the most well-known such problem is their conjecture that $\alpha_1(G) + \tau_1(G) \leq |V(G)|^2/4$, which was proved by Norin and Sun. We consider three other problems within this theme (two introduced by Erd\H{o}s, Gallai, and Tuza, another by Norin and Sun), all of which request an upper bound either on $\min\{\alpha_1(G), \tau_1(G)\}$ or on $\alpha_1(G) + k\tau_1(G)$ for some constant $k$, and prove the existence of graphs for which these quantities are "large".