On the minimal density of triangles in graphs

  • Authors:

  • Alexander a. Razborov

  • Affiliations:

  • Institute for advanced study, princeton, usa and steklov mathematical institute, moscow, russia (e-mail: razborov@ias.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2008

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Abstract

For a fixed ρ ∈ [0, 1], what is (asymptotically) the minimal possible density g3(ρ) of triangles in a graph with edge density ρ? We completely solve this problem by proving that $$ g_3(\rho) =\frac{(t-1)\ofb{t-2\sqrt{t(t-\rho(t+1))}}\ofb{t+\sqrt{t(t-\rho(t+1))}}^2}{t^2(t+1)^2},$$ where $t\df \lfloor 1/(1-\rho)\rfloor$ is the integer such that $\rho\in\bigl[ 1-\frac 1t,1-\frac 1{t+1}\bigr]$.