A problem of Erd\H{o}s and Rousseau. The constant $50$ would be best possible as witnessed by a blow-up of $C_5$ or the Petersen graph.

Erd\H{o}s, Faudree, Rousseau, and Schelp \cite{EFRS94} proved that this is true with $50$ replaced by $16$. More generally, they prove that, for any $0<\alpha<1$, if every set of $\geq \alpha n$ vertices contains $>\alpha^3n^2/2$ edges then $G$ contains a triangle.

Krivelevich \cite{Kr95} has proved this with $n/2$ replaced by $3n/5$ (and $50$ replaced by $25$).

Keevash and Sudakov \cite{KeSu06} have proved this under the additional assumption that either $G$ has at most $n^2/12$ edges, or that $G$ has at least $n^2/5$ edges. Norin and Yepremyan \cite{NoYe15} proved that this is true if $G$ has at least $(1/5-c)n^2$ edges, for some constant $c>0$.

Razborov \cite{Ra22} proved this is true if $\frac{1}{50}$ is replaced by $\frac{27}{1024}$.

See also the entry in the graphs problem collection.
https://mathweb.ucsd.edu/~erdosproblems/erdos/newproblems/LocalEdgeDensities.html


References


[EFRS94] Erd\H{o}s, P. and Faudree, R. J. and Rousseau, C. C. and
Schelp, R. H., A local density condition for triangles. Discrete Math. (1994), 153--161.

[KeSu06] Keevash, Peter and Sudakov, Benny, Sparse halves in triangle-free graphs. J. Combin. Theory Ser. B (2006), 614-620.

[Kr95] Krivelevich, Michael, On the edge distribution in triangle-free graphs. J. Combin. Theory Ser. B (1995), 245-260.

[NoYe15] Norin, Sergey and Yepremyan, Liana, Sparse halves in dense triangle-free graphs. J. Combin. Theory Ser. B (2015), 1--25.

[Ra22] Razborov, A. A., More about sparse halves in triangle-free graphs. Mat. Sb. (2022), 119--140.