Editing Turán's theorem (section)

==Proofs==
{{harvtxt|Aigner|Ziegler|2018}} list five different proofs of Turán's theorem.{{r|az}}
Many of the proofs involve reducing to the case where the graph is a complete [[multipartite graph]], and showing that the number of edges is maximized when there are <math>r</math> parts of size as close as possible to equal.

=== Induction ===
[[File:Turán-Induct-r=3.png|thumb|256x256px|(Induction on n) An example of sets <math>A</math> and <math>B</math> for <math>r=3</math>.]]
[[File:Turán-Erdős-Replacement.png|thumb|299x299px|(Maximal Degree Vertex) Deleting edges within <math>A</math> and drawing edges between <math>A</math> and <math>B</math>.]]
This was Turán's original proof. Take a <math>K_{r+1}</math>-free graph on <math>n</math> vertices with the maximal number of edges. Find a <math>K_r</math> (which exists by maximality), and partition the vertices into the set <math>A</math> of the <math>r</math> vertices in the <math>K_r</math> and the set <math>B</math> of the <math>n-r</math> other vertices.

Now, one can bound edges above as follows:

* There are exactly <math>\binom{r}{2}</math> edges within <math>A</math>.
* There are at most <math>(r-1)|B|=(r-1)(n-r)</math> edges between <math>A</math> and <math>B</math>, since no vertex in <math>B</math> can connect to all of <math>A</math>.
* The number of edges within <math>B</math> is at most the number of edges of <math>T(n-r,r)</math> by the inductive hypothesis.

Adding these bounds gives the result.{{r|turan|az}}

=== Maximal Degree Vertex ===
This proof is due to [[Paul Erdős]]. Take the vertex <math>v</math> of largest degree. Consider the set <math>A</math> of vertices not adjacent to <math>v</math> and the set <math>B</math> of vertices adjacent to <math>v</math>.

Now, delete all edges within <math>A</math> and draw all edges between <math>A</math> and <math>B</math>. This increases the number of edges by our maximality assumption and keeps the graph <math>K_{r+1}</math>-free. Now, <math>B</math> is <math>K_r</math>-free, so the same argument can be repeated on <math>B</math>.

Repeating this argument eventually produces a graph in the same form as a [[Turán graph]], which is a collection of independent sets, with edges between each two vertices from different independent sets. A simple calculation shows that the number of edges of this graph is maximized when all independent set sizes are as close to equal as possible.{{r|az|erdos}}

=== Complete Multipartite Optimization ===
This proof, as well as the Zykov Symmetrization proof, involve reducing to the case where the graph is a complete [[multipartite graph]], and showing that the number of edges is maximized when there are <math>r</math> independent sets of size as close as possible to equal. This step can be done as follows:

Let <math>S_1, S_2, \ldots, S_r</math> be the independent sets of the multipartite graph. Since two vertices have an edge between them if and only if they are not in the same independent set, the number of edges is

<math display="block">\sum_{i \neq j} \left|S_i\right|\left|S_j\right|=\frac{1}{2}\left(n^2-\sum_{i} \left|S_i\right|^2\right),</math>

where the left hand side follows from direct counting, and the right hand side follows from complementary counting. To show the <math>\left(1-\frac{1}{r}\right)\frac{n^2}{2}</math> bound, applying the [[Cauchy–Schwarz inequality]] to the <math display="inline">\sum\limits_i\left|S_i\right|^2</math> term on the right hand side suffices, since <math display="inline">\sum\limits_i\left|S_i\right|=n</math>.

To prove the Turán Graph is optimal, one can argue that no two <math>S_i</math> differ by more than one in size. In particular, supposing that we have <math>\left|S_i\right| \geq \left|S_j\right|+2</math> for some <math>i \neq j</math>, moving one vertex from <math>S_j</math> to <math>S_i</math> (and adjusting edges accordingly) would increase the value of the sum. This can be seen by examining the changes to either side of the above expression for the number of edges, or by noting that the degree of the moved vertex increases.

=== Lagrangian ===
This proof is due to {{harvtxt|Motzkin|Straus|1965}}. They begin by considering a <math>K_{r+1}</math> free graph with vertices labelled <math>1,2,\ldots,n</math>, and considering maximizing the function<math display="block">f(x_1,x_2,\ldots,x_n)=\sum_{i,j\ \text{adjacent}} x_ix_j</math>over all nonnegative <math>x_1,x_2,\ldots,x_n</math> with sum <math>1</math>. This function is known as the [[Homomorphism density#Lagrangian|Lagrangian]] of the graph and its edges.

The idea behind their proof is that if <math>x_i,x_j</math> are both nonzero while <math>i,j</math> are not adjacent in the graph, the function<math display="block">f(x_1,\ldots,x_i-t,\ldots,x_j+t,\ldots,x_n)</math>is linear in <math>t</math>. Hence, one can either replace <math>(x_i,x_j)</math> with either <math>(x_i+x_j,0)</math> or <math>(0,x_i+x_j)</math> without decreasing the value of the function. Hence, there is a point with at most <math>r</math> nonzero variables where the function is maximized.


Now, the [[Cauchy–Schwarz inequality]] gives that the maximal value is at most  <math>\frac{1}{2}\left(1-\frac{1}{r}\right)</math>. Plugging in <math>x_i=\frac{1}{n}</math> for all <math>i</math> gives that the maximal value is at least <math>\frac{|E|}{n^2}</math>, giving the desired bound.{{r|az|ms}}

=== 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.]]

=== Zykov Symmetrization ===
Aigner and Ziegler call the final one of their five proofs "the most beautiful of them all". Its origins are unclear, but the approach is often referred to as Zykov Symmetrization as it was used in Zykov's proof of a generalization of Turán's Theorem {{r|zykov}}. This proof goes by taking a <math>K_{r+1}</math>-free graph, and applying steps to make it more similar to the Turán Graph while increasing edge count.

In particular, given a <math>K_{r+1}</math>-free graph, the following steps are applied:

* If <math>u,v</math> are non-adjacent vertices and <math>u</math> has a higher degree than <math>v</math>, replace <math>v</math> with a copy of <math>u</math>. Repeat this until all non-adjacent vertices have the same degree.
* If <math>u,v,w</math> are vertices with <math>u,v</math> and <math>v,w
</math> non-adjacent but <math>u,w</math> adjacent, then replace both <math>u</math> and <math>w
</math> with copies of <math>v
</math>.

All of these steps keep the graph <math>K_{r+1}</math> free while increasing the number of edges.

Now, non-adjacency forms an [[equivalence relation]]. The [[equivalence class]]es give that any maximal graph the same form as a Turán graph. As in the maximal degree vertex proof, a simple calculation shows that the number of edges is maximized when all independent set sizes are as close to equal as possible.{{r|az}}