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On the minimal density of triangles in graphs. (English) Zbl 1170.05036
Summary: For a fixed $$\rho\in[0,1]$$, what is (asymptotically) the minimal possible density $$g_3(\rho)$$ of triangles in a graph with edge density $$\rho$$? We completely solve this problem by proving that
$g_3(\rho)= \frac{(t-1)\big(t-2\sqrt{t(t-\rho(t+1))}\big) \big(t+\sqrt{t(t-\rho(t+1))}\big)^2}{t^2(t+1)^2},$ where $$t \overset\text{def}= \lfloor 1/(1-\rho)\rfloor$$ is the integer such that $$\rho\in[1-\frac 1t,1-\frac 1{t+1}]$$.

##### MSC:
 05C35 Extremal problems in graph theory 05E10 Combinatorial aspects of representation theory
##### Keywords:
graph; density of triangles
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##### References:
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