Summary: Let \({\mathcal G}(n;e)\) denote the class of graphs on n vertices and e edges. Define \(f(n,e)=\min \max \{\sum^{3}_{i=1}d(v_ i):\{v_ 1,v_ 2,v_ 3\}\) induces a triangle in \(G\}\) where the maximum is taken over all triangles in the graph G and the minimum is taken over all G in \({\mathcal G}(n;e)\). From Turán’s theorem, \(f(n,e)=0\) if \(e\leq n^ 2/4\); otherwise \(f(n,e)>0\). B. Bollobás and P. Erdős [Unsolved problems, Proc. 5th British Combinatorial Conference, Utilitas Mathematica Publishing Inc., Winnipeg, 678-680 (1975)] asked to determine the function f(n,e) for \(e>n^ 2/4\). C. S. Edwards [Bull. Lond. Math. Soc. 9, 203-208 (1977; Zbl 0357.05058)] proved that f(n,e)\(\geq 6e/n\) for \(e\geq n^ 2/3\). In this paper, we consider the remaining case, namely, \(n^ 2/4<e<n^ 2/3\). A construction described in [P. Erdős and R. Laskar, Combinatorics, graph theory and computing, Proc. 16th Southeast. Conf., Boca Raton/Fla. 1985, Congr. Numerantium 48, 81-86 (1985; Zbl 0647.05034)] shows that \(f(n,e)<4\sqrt{3e}-2n+5.\) We prove that \(f(n,e)\geq 21e/4n.\) In particular, \(f(n,\lfloor n^ 2/4\rfloor +1)>21n/16,\) which improves a result of Erdős and Laskar [op. cit.] that \(f(n,\lfloor n^ 2/4\rfloor +1)>(1+\epsilon)n\) for some positive constant \(\epsilon\). Furthermore, if \(e\geq 0.26n^ 2\), we obtain a better result.