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

 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?

 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.

 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.

 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)

 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 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

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

What graphs has many cliques and anticliques? Example: r=s=3.  First guess: A clique on some fraction of the vertices  Second guess: Complements of these graphs t1-t

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.

 Stability: such that every sufficiently large graph G with is close to the extremal graph.  Max-min: where

 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.

 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.

 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.

 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.

 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 A_1 A_2 A_3 A_4 A_2k-1 A_2k

 The densities are (upto o(1)): 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.

 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.

 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,above 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. A_1 A_2 A_3 A_4 A_2k-1 A_2k A_1 U A_3

 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.

 Define the following matrices:  If x_3>0 and (x,y) is non-degenerate then B is positive definite.B is positive definite.  For, let x’ be defined by

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

 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. Stability

?

 For l≤m,  Hence, and