Which graphs are extremal Lszl Lovsz Etvs Lornd

Which graphs are extremal? László Lovász Eötvös Loránd University, Budapest Joint work with Balázs Szegedy

Types of extremal graphs Given: # of nodes n, # edges m, minimize # triangles. Triangles

Types of extremal graphs Triangles Kruskal-Katona 1 Razborov 2006 Fisher Goodman 0 1/2 2/3 3/4 1 Bollobás Mantel-Turán Lovász-Simonovits

Types of extremal graphs Triangles Given: # of nodes n, # edges m, minimize # triangles. Mantel (1907): Goodman (1959): extremal graph is Razborov (2006): extremal graph is

Types of extremal graphs Nonbipartite excluded subgraphs Given: # of nodes n, excluded subgraphs F 1, . . . , Fm, minimize # edges. Erdős-Stone-Simonovits: Asymptotic extreme graph: , where

Types of extremal graphs Quadrilaterals Given: # of nodes n, # edges m, minimize # quadrilaterals. Asymptotically: Asymptotic extreme graph: random

Limits of dense graph sequences Borgs, Chayes, L, Sós, B. Szegedy, Vesztergombi Probability that random map V(G) V(H) is a hom convergent: for every simple graph is convergent

The limit object as a function graphons

The limit object as a function Example 1: Adjacency matrix of graph G: Examples Associated function WG:

The limit object as a function Examples Stepfunction: Stepfunctions finite graphs with node and edgeweights

The limit object as a function Examples Example 2: t(F, W)= 2 -|E(F)| # of eulerian orientations of F

The limit object as a function

Summary of main facts about graph limits For every convergent graph sequence (Gn) there is a graphon such that Conversely, for every graphon W there is a graph sequence (Gn) such that W is essentially unique (up to measure-preserving transformation).

Extremal graphons Extremal graphon problem: Given find subject to

Finite forcing Graphon W is finitely forcible: Every finitely forcible graphon is extremal: minimize Every unique extremal graphon is finitely forcible. ? ? ? Every extremal graph problem has a finitely forcible extremal graphon ? ? ?

Finite forcing Examples Goodman 1/2 Graham. Chung. Wilson

Finite forcing Finitely expressible properties d-regular graphon: d-regular

Finite forcing Finitely expressible properties W is 0 -1 valued, and can be rearranged to be monotone decreasing in both variables “W is 0 -1 valued” is not finitely expressible. W is 0 -1 valued

Finite forcing Stepfunctions Every stepfunction is finitely forcible Is the converse true? L – T. Sós

Finite forcing One-variable analogue (F connected) The t(F, W ) determine W up to measure preserving transformation The M(k, f ) determine f up to measure preserving transformation t(F 1, W ), . . . t(Fm, W ) are independent. M(k 1, f ), . . . M(km, f ) are independent. Characterizations: inclusion-exclusion inequalities, semidefiniteness, . . .

Finite forcing One-variable analogue W stepfunction finite number of f stepfunction finite number of subgraph densities that determine it. moments that determine it. ? f not a stepfunction k 1, . . . , km stepfunction g such that

Finite forcing Signs of polynomials p monotone decreasing symmetric polynomial finitely forcible ?

Finite forcing Signs of polynomials, proof sketch S p(x, y)=0

Finite forcing 2 -dimensional graphons Is the following graphon finitely forcible? angle <π/2

Finite forcing UC graphons Union-complement graph (UC-graphs, cographs): Constructed from single nodes by disjoint union and complementation G is UC no induced P 4 union and complement single node

Finite forcing UC graphons UC-graphon: not connected probability measure on paths UC-graphon is a stepfunction tree is finite . . . connected

Finite forcing Regular UC graphons UC-graphon is regular tree is locally finite For every locally finite tree with outdegrees ≥ 2 there is a unique regular UC-graphon. d-regular UC-graphon is a stepfunction d is rational 0 ≤ d ≤ 1 d-regular UC-graphon

Finite forcing Regular UC graphons W is not a stepfunction

Finite forcing Non-forcible graphons W(x, y) is a polynomial not finitely forcible ? ? Every finitely forcible function has a finite range. ? ? (stepfunction in a weaker sense)