Editing Turán's theorem (section)

=== Maximizing Other Quantities ===
Another natural extension of Turán's theorem is the following question: if a graph has no <math>K_{r+1}</math>s, how many copies of <math>K_{a}</math> can it have? Turán's theorem is the case where <math>a=2</math>. Zykov's Theorem answers this question:<blockquote>(Zykov's Theorem) The graph on <math>n</math> vertices with no <math>K_{r+1}</math>s and the largest possible number of <math>K_{a}</math>s is the Turán graph <math>T(n,r)</math></blockquote>This was first shown by Zykov (1949) using Zykov Symmetrization{{r|turan|az}}. Since the Turán Graph contains <math>r</math> parts with size around <math>\frac{n}{r}</math>, the number of <math>K_{a}</math>s in <math>T(n,r)</math> is around <math>\binom{r}{a}\left(\frac{n}{r}\right)^a</math>.

A paper by Alon and Shikhelman in 2016 gives the following generalization, which is similar to the Erdos-Stone generalization of Turán's theorem:<blockquote>(Alon-Shikhelman, 2016) Let <math>H</math> be a graph with chromatic number <math>\chi(H)>a</math>. The largest possible number of <math>K_{a}</math>s in a graph with no copy of <math>H</math> is

<math>(1+o(1))\binom{\chi(H)-1}{a}\left(\frac{n}{\chi(H)-1}\right)^a.</math>{{r|alonshik}}</blockquote>As in Erdős–Stone, the Turán graph <math>T(n,\chi(H)-1)</math> attains the desired number of copies of <math>K_{a}</math>.