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Turán's theorem
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=== Probabilistic Method === The key claim in this proof was independently found by Caro and Wei. This proof is due to [[Noga Alon]] and [[Joel Spencer]], from their book ''The Probabilistic Method''. The proof shows that every graph with degrees <math>d_1,d_2,\ldots,d_n</math> has an [[Independent set (graph theory)|independent set]] of size at least<math display="block">S=\frac{1}{d_1+1}+\frac{1}{d_2+1}+\cdots+\frac{1}{d_n+1}.</math>The proof attempts to find such an independent set as follows: * Consider a [[random permutation]] of the vertices of a <math>K_{r+1}</math>-free graph * Select every vertex that is adjacent to none of the vertices before it. A vertex of degree <math>d</math> is included in this with probability <math>\frac{1}{d+1}</math>, so this process gives an average of <math>S</math> vertices in the chosen set. [[File:Turán-Zykov-Step-1.png|thumb|375x375px|(Zykov Symmetrization) Example of first step.]] Applying this fact to the [[complement graph]] and bounding the size of the chosen set using the Cauchy–Schwarz inequality proves Turán's theorem.{{r|az}} See {{slink|Method_of_conditional_probabilities|Turán's_theorem}} for more. [[File:Turán-Zykov-Step-2.png|thumb|375x375px|(Zykov Symmetrization) Example of second step.]]
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