Densities of cliques and independent sets in graphs

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.

Local profiles of graphs 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 two cliques § 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)

A clique and an anticlique § 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 (II) § § 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

Many cliques and anticliques How big can both d(Ks; G) and d(Kr; G) be?

Many cliques and anticliques What graphs has many cliques and anticliques? § First guess: A clique on some fraction of the vertices Example: r=s=3. 1 0. 9 0. 8 t 1 -t 0. 7 0. 6 0. 5 § Second guess: Complements of these graphs 0. 4 0. 3 0. 2 0. 1 0 0 0. 2 0. 4 0. 6 0. 8 1

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. §

Shifting in a nutshell § 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 cliques § § § 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.

Threshold graphs § 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. §

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. §

Threshold graphs § Every threshold graph G can be encoded as a point in A_1 A_2 A_3 A_4 A_2 k-1 A_2 k

densities in threshold graphs § The densities are (upto o(1)): A_1 A_2 A_3 A_4 A_2 k-1 A_2 k

Optimization problem § § 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.

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. §

Technical lemma § 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_1 A_3 A_2 A_4 a suffix. A_1 U A_3 A_2 k-1 A_2 k

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.

Remarks § For the max-min theorem: Consider 1 0. 9 (a=b=1). 0. 8 0. 7 § For r=s=3, Goodman inequality and our bound completely determine the set 0. 6 0. 5 0. 4 0. 3 § Stability – obtained using Keevash’s stable Kruskal-Katona theorem. 0. 2 0. 1 0 0 0. 2 0. 4 0. 6 0. 8 1

The End ?

§ For l≤m, § Hence, and