Densities of cliques and independent sets in graphs

Yuval Peled, HUJI

Joint work with Nati Linial, Benny Sudakov, Hao Huang and Humberto Naves.

High level motivation

How can we study large graphs?

Approach: Sample small sets of vertices and examine the induced subgraphs.

What graph properties can be inferred from its local profile?

What are the possible local profiles of large graphs?

Local profiles of graphs

What are the possible local profiles of (large) graphs?

For graphs H,G, we denote by d(H;G) the induced density of H in G, i.e.

d(H;G):= The probability that |H| random vertices in G induce a copy of H.

Definition: Given a family of graphs, is the set of all such that , a sequence of graphs with

and

Problem: Characterize this set.

Local profiles of graphs

Characterizing seems to be a hard task: A mathematical perspective:

Many hard problems fall into this framework. E.g. for t=1, the problem is equivalent to computing

the inducibility of graph,

a parameter known only for a handful of graphs. A computational perspective:

[Hatami, Norine 11’]: Satisfiability of linear inequalities in is undecidable.

Local profiles of graphs

The case of two cliques is already of interest: Turan’s Theorem:

Kruskal-Katona Theorem: (r<s)

Minimize subject to this constraint? much harder: solved only recently for r=2:Razborov 08’ (s=3), Nikiforov 11’ (s=4(, Reiher (arbitrary s)

Local profiles of two cliques

Motivation - quantitative versions of Ramsey’s theorem: Investigate distributions of monochromatic cliques in a

red/blue coloring of the complete graph. Goodman’s inequality:

The minimum is attained by G(n,½), conjectured by Erdos to minimize

for every r. Refuted by Thomasson for every r>3.

A clique and an anticlique

A consequence from Goodman’s inequality:

[Franek-Rodl 93’] The analog of this is false for r=4, by a blow up of the following graph: V = {0,1}^13, v~u iff dist(v,u) ∈ {1,4,5,8,9,11}

Fundamental open problem: Find graphs with few cliques and anticliques.

We are interested in the other side of

A clique and an anticlique (II)

How big can both d(Ks;G) and

d(Kr;G) be?

Many cliques and anticliques

What graphs has many cliques and anticliques?Example: r=s=3.

Many cliques and anticliques

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First guess:A clique on some fraction of the vertices

Second guess:Complements of these graphs

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t 1-t

Main theorem

Let r, s > 2. Suppose that and let q be

the unique root in [0,1] of

Then,

Namely, given the maximum of is

attained in one of two graphs: a clique on a

fraction of the vertices, or the complement of

such graph.

More theorems Stability: such that every

sufficiently large graph G with

is close to the extremal graph. Max-min:

where

Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold

graphs as an optimization problem.III. Characterize the solutions of the

optimization problem.

Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold

graphs as an optimization problem.III. Characterize the solutions of the

optimization problem.

Given a graph G and vertices u,v the shift of G from u to v is defined by the rule: Every other vertex w with w~u and w≁v gets

disconnected from u and connected to v. A graph G with V=[n] is said to be shifted if

for every i<j the shift of G from j to i does not change G.

Fact: Every graph can be made shifted by a finite number of shifting operations.

Shifting in a nutshell

Lemma: Shifting does not decrease the number of s-cliques in the graph.

Proof: Consider the shift from j to i. If a subset C of V forms a clique in G and not in the shifted graph S(G), then C \ {j} U {i} forms a clique in S(G) and not in G.

Cor: By symmetry, shifting does not decrease the number of r-anticliques.

Shifting cliques

Def: A graph is called a threshold graph if there is an order on the vertices, such that every vertex is adjacent to either all or none of its predecessors.

Lemma: A shifted graph is a threshold graph. Proof: Consider the following order:

Cor: The extremal graph is a threshold graph.

Threshold graphs

Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold

graphs as an optimization problem.III. Characterize the solutions of the

optimization problem.

Every threshold graph G can be encoded as a point in

Threshold graphs

A_1 A_2 A_3 A_4 A_2k-1 A_2k

The densities are (upto o(1)):

densities in threshold graphs

A_1 A_2 A_3 A_4 A_2k-1 A_2k

The new form of our optimization problem is:

We need to prove that every maximum is either supported on x_1,y_1 or on y_1,x_2.

Optimization problem

It suffices to show that for every a,b>0, the maximum of

is either supported on x_1,y_1 or on y_1,x_2.

Why? For both problems have the same set of

maximum points.

Optimization problem

Proof of main theorem Strategy:I. Reduce the problem to threshold graphs.II. Reformulate the problem for threshold

graphs as an optimization problem.III. Characterize the solutions of the

optimization problem.

Let k,r,s≥2 be integers, a,b>0 reals, and

the polynomials defined above. Then,every non-degenerate maximum of

is either supported on x_1,y_1 or on y_1,x_2.

(x,y) is non-degenerate if the zeros in the sequence (y_1,x_2,y_3,…,x_k,y_k) form a suffix.

Technical lemma

A_1 A_2 A_3 A_4 A_2k-1 A_2kA_1 U A_3

Proof Let (x,y) be a non-degenerate maximum of

f:

, otherwise we can increase f by a perturbation that increases the smaller element.

WLOG x_1>0, otherwise x exchange roles with y, and p with q (by looking at the complement graph).

We show that x_3=y_2=x_2=0.

Proof (x_3=0) Define the following matrices:

If x_3>0 and (x,y) is non-degenerate then B is positive definite.

For , let x’ be defined by

Proof (x_3=0) (II)

Then,

If A is singular – choose Av=0, v≠0. If A is invertible – choose

Proof (x_3=0) (III) Hence,

contradicting the maximality of f(x,y).

Proving y_2=0, x_2=0 is done with similar methods.

For the max-min theorem: Consider

(a=b=1).

For r=s=3, Goodman inequality and our bound completely determine the set

Stability – obtained using Keevash’s stable Kruskal-Katona theorem.

Remarks

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The End

For l≤m,

Hence,

and