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The triangle covering number of a graph is the minimum number of verticesthat hit all triangles. Given positive integers $s,t$ and an $n$-vertex graph$G$ with $\lfloor n^2/4 \rfloor +t$ edges and triangle covering number $s$, wedetermine (for large $n$) sharp bounds on the minimum number of triangles in$G$ and also describe the extremal constructions. Similar results are provedfor cliques of larger size and color critical graphs.  This extends classical work of Rademacher, Erd\H os, and Lov\'asz-Simonovitswhose results apply only to $s \le t$. Our results also address two conjecturesof Xiao and Katona. We prove one of them and give a counterexample and prove amodified version of the other conjecture.

On a generalized Erd\H{o}s-Rademacher problem