Forcing triangles with local edge densities

The following problem[2] relates the edge density of a graph to the containment of triangles:

A conjecture on local edge density and triangles (proposed by Erdös and Rousseau[2])

If each set of $$\lfloor n/2 \rfloor$$ vertices in a graph $$G$$ with $$n$$ vertices spans more than $$n^2/50$$ edges, then $$G$$ contains a triangle.

A more general but slightly weaker version was proved [4] which asserts that for all sufficiently large $$n$$, a graph on $$n$$ vertices in which each set of $$\lfloor \alpha n \rfloor$$ vertices spans at least $$\beta n^2$$ edges must contain a triangle if $$\alpha \geq .648$$ and $$\beta > (2 \alpha -1)/4$$. Krivelevich [3] showed that a graph on $$n$$ vertices in which each set of $$\lfloor \alpha n \rfloor$$ vertices spans at least $$\beta n^2$$ edges must contain a triangle if $$\alpha \geq .6$$ and $$\beta > (5 \alpha -2)/25$$.

Chung and Graham [1] made the following related conjecture:

Conjecture

Let $$b_t(n)$$ denote the maximum number of edges induced by any set of $$\lfloor n/2 \rfloor$$ vertices in the Turán graph on $$n$$ vertices for $$K_t$$. If each set of $$\lfloor n/2 \rfloor$$ vertices in a graph with $$n$$ vertices spans more than $$b_t(n)$$ edges, then $$G$$ contains a $$K_t$$.

Bibliography
1 F. R. K. Chung and R. L. Graham. On graphs not containing prescribed induced subgraphs. A Tribute to Paul Erdös, (eds. A. Baker, B. Bollobás, and A. Hajnal), Cambridge University Press, Cambridge, (1990), 111-120.

2 P. Erdös and C. C. Rousseau. The size Ramsey number of a complete bipartite graph. Discrete Math. 113 (1993) no. 1-3, 259-262.

3 M. Krivelevich. On the edge distribution in triangle-free graphs. J. Comb. Theory (B) 63 (1995), 245-260.

4 P. Erdös, R. Faudree, C. C. Rousseau and R. H. Schelp, A local density condition for triangles, Discrete Math., Graph theory and applications (Hakone, 1990), Discrete Math. 127 (1994), 153-161.