Undecidability of Linear Inequalities Between Graph Homomorphism Densities
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1 Undecidability of Linear Inequalities Between Graph Homomorphism Densities Hamed Hatami joint work with Sergey Norin School of Computer Science McGill University December 4, 2013 Hamed Hatami (McGill University) December 4, / 43
2 Introduction Hamed Hatami (McGill University) December 4, / 43
3 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Hamed Hatami (McGill University) December 4, / 43
4 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Few techniques are very common (Induction, Cauchy-Schwarz,... ). Hamed Hatami (McGill University) December 4, / 43
5 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Few techniques are very common (Induction, Cauchy-Schwarz,... ). Recently, there has been several developments explaining this: Freedman, Lovász, Schrijver, Szegedy, Razborov, Chayes, Borgs, Sös, Vesztergombi,... Hamed Hatami (McGill University) December 4, / 43
6 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Few techniques are very common (Induction, Cauchy-Schwarz,... ). Recently, there has been several developments explaining this: Freedman, Lovász, Schrijver, Szegedy, Razborov, Chayes, Borgs, Sös, Vesztergombi,... Discovery of rich algebraic structure underlying many of these techniques. Hamed Hatami (McGill University) December 4, / 43
7 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Few techniques are very common (Induction, Cauchy-Schwarz,... ). Recently, there has been several developments explaining this: Freedman, Lovász, Schrijver, Szegedy, Razborov, Chayes, Borgs, Sös, Vesztergombi,... Discovery of rich algebraic structure underlying many of these techniques. Neater proofs with no low-order terms. Hamed Hatami (McGill University) December 4, / 43
8 Asymptotic extremal graph theory has been studied for more than a century (Mantel 1908). Few techniques are very common (Induction, Cauchy-Schwarz,... ). Recently, there has been several developments explaining this: Freedman, Lovász, Schrijver, Szegedy, Razborov, Chayes, Borgs, Sös, Vesztergombi,... Discovery of rich algebraic structure underlying many of these techniques. Neater proofs with no low-order terms. Methods for applying these techniques in semi-automatic ways. Hamed Hatami (McGill University) December 4, / 43
9 Density of H in G number of copies of H in G ). ( V (G) V (H) Hamed Hatami (McGill University) December 4, / 43
10 Density of H in G number of copies of H in G ). ( V (G) V (H) We can think of these densities as moments of the graph G. Hamed Hatami (McGill University) December 4, / 43
11 Density of H in G number of copies of H in G ). ( V (G) V (H) We can think of these densities as moments of the graph G. Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between subgraph densities. Hamed Hatami (McGill University) December 4, / 43
12 Theorem (Freedman, Lovász, Schrijver 2007) Every such inequality follows from the positive semi-definiteness of a certain infinite matrix. Hamed Hatami (McGill University) December 4, / 43
13 Theorem (Freedman, Lovász, Schrijver 2007) Every such inequality follows from the positive semi-definiteness of a certain infinite matrix. Equivalently (possibly infinitely many) applications of Cauchy-Schwarz. Hamed Hatami (McGill University) December 4, / 43
14 Theorem (Freedman, Lovász, Schrijver 2007) Every such inequality follows from the positive semi-definiteness of a certain infinite matrix. Equivalently (possibly infinitely many) applications of Cauchy-Schwarz. Razborov s flag algebras A formal calculus capturing many standard arguments (induction, Cauchy-Schwarz,...) in the area. Hamed Hatami (McGill University) December 4, / 43
15 Applications Automatic methods for proving theorems (based on SDP): Hamed Hatami (McGill University) December 4, / 43
16 Applications Automatic methods for proving theorems (based on SDP): Razborov: Significant improvement over the known bounds on Turán s Hypergraph Problem. Hamed Hatami (McGill University) December 4, / 43
17 Applications Automatic methods for proving theorems (based on SDP): Razborov: Significant improvement over the known bounds on Turán s Hypergraph Problem. HH, Hladky, Kral, Norin, Razborov: A question of Sidorenko and Jagger, Št ovíček and Thomason. Hamed Hatami (McGill University) December 4, / 43
18 Applications Automatic methods for proving theorems (based on SDP): Razborov: Significant improvement over the known bounds on Turán s Hypergraph Problem. HH, Hladky, Kral, Norin, Razborov: A question of Sidorenko and Jagger, Št ovíček and Thomason. HH, Hladky, Kral, Norin, Razborov: A conjecture of Erdös. Hamed Hatami (McGill University) December 4, / 43
19 Applications SDP methods + thinking: Hamed Hatami (McGill University) December 4, / 43
20 Applications SDP methods + thinking: Razborov: Minimal density of triangles, given an edge density. Hamed Hatami (McGill University) December 4, / 43
21 Applications SDP methods + thinking: Razborov: Minimal density of triangles, given an edge density. Razborov: Turán s hypergraph problem under mild extra conditions. Hamed Hatami (McGill University) December 4, / 43
22 Applications SDP methods + thinking: Razborov: Minimal density of triangles, given an edge density. Razborov: Turán s hypergraph problem under mild extra conditions. other conjectures of Erdös, crossing number of complete bipartite graphs, etc. Hamed Hatami (McGill University) December 4, / 43
23 How far can we go? Is asymptotic extremal graph theory trivial? Is lack of enough computational power the only barrier? Hamed Hatami (McGill University) December 4, / 43
24 How far can we go? Is asymptotic extremal graph theory trivial? Is lack of enough computational power the only barrier? Question (Razborov) Can every true algebraic inequality between subgraph densities be proved using a finite amount of manipulation with subgraph densities of finitely many graphs? Hamed Hatami (McGill University) December 4, / 43
25 How far can we go? Is asymptotic extremal graph theory trivial? Is lack of enough computational power the only barrier? Question (Razborov) Can every true algebraic inequality between subgraph densities be proved using a finite amount of manipulation with subgraph densities of finitely many graphs? HH-Norine 2011 The answer is negative in a strong sense. Hamed Hatami (McGill University) December 4, / 43
26 Formal definitions Hamed Hatami (McGill University) December 4, / 43
27 Extremal graph theory Studies the relations between the number of occurrences of different subgraphs in a graph G. Hamed Hatami (McGill University) December 4, / 43
28 Extremal graph theory Studies the relations between the number of occurrences of different subgraphs in a graph G. Equivalently one can study the relations between the homomorphism densities. Hamed Hatami (McGill University) December 4, / 43
29 Homomorphism Density Definition Map the vertices of H to the vertices of G independently at random. H Hamed Hatami (McGill University) December 4, / 43
30 Homomorphism Density Definition Map the vertices of H to the vertices of G independently at random. t H (G) := Pr[edges go to edges]. H Hamed Hatami (McGill University) December 4, / 43
31 Definition A map f : H G is called a homomorphism if it maps edges to edges. Hamed Hatami (McGill University) December 4, / 43
32 Definition A map f : H G is called a homomorphism if it maps edges to edges. t H (G) = Pr[f : H G is a homomorphism]. Hamed Hatami (McGill University) December 4, / 43
33 Asymptotically t H ( ) and subgraph densities are equivalent. Hamed Hatami (McGill University) December 4, / 43
34 Asymptotically t H ( ) and subgraph densities are equivalent. The functions t H have nice algebraic structures: t H1 H 2 (G) = t H1 (G)t H2 (G). H1 G H2 Hamed Hatami (McGill University) December 4, / 43
35 Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between homomorphism densities. Hamed Hatami (McGill University) December 4, / 43
36 Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between homomorphism densities. Example (Goodman s bound 1959) t K3 (G) 2t K2 (G) 2 t K2 (G). Hamed Hatami (McGill University) December 4, / 43
37 Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between homomorphism densities. Example (Goodman s bound 1959) t K3 (G) 2t K2 (G) 2 t K2 (G). Every such inequality can be turned to a linear inequality: a 1 t H1 (G) a m t Hm (G) 0. Hamed Hatami (McGill University) December 4, / 43
38 Many fundamental theorems in extremal graph theory can be expressed as algebraic inequalities between homomorphism densities. Example (Goodman s bound 1959) t K3 (G) 2t K2 (G) 2 t K2 (G). Every such inequality can be turned to a linear inequality: a 1 t H1 (G) a m t Hm (G) 0. Example (Goodman s bound 1959) t K3 (G) 2t K2 K 2 (G) + t K2 (G) 0. Hamed Hatami (McGill University) December 4, / 43
39 Algebra of Partially labeled graphs Hamed Hatami (McGill University) December 4, / 43
40 Definition A partially labeled graph is a graph in which some vertices are labeled by distinct natural numbers. Example 2 3 Hamed Hatami (McGill University) December 4, / 43
41 Definition A partially labeled graph is a graph in which some vertices are labeled by distinct natural numbers. Recall t H (G) := Pr [f : H G is a homomorphism]. Hamed Hatami (McGill University) December 4, / 43
42 Definition A partially labeled graph is a graph in which some vertices are labeled by distinct natural numbers. Recall t H (G) := Pr [f : H G is a homomorphism]. Definition Let H be partially labeled with labels L. For φ : L G, define t H,φ (G) := Pr [f : H G is a hom. f L = φ]. Hamed Hatami (McGill University) December 4, / 43
43 Definition Let H be partially labeled with labels L. For φ : L G, define t H,φ (G) := Pr [f : H G is a hom. f L = φ]. Example G H 1 φ t H,φ (G) = 3 6 = 1 2. Hamed Hatami (McGill University) December 4, / 43
44 Example G G H [[H]] 1 φ Definition Let [H] be H with no labels. Hamed Hatami (McGill University) December 4, / 43
45 Example G G H [[H]] 1 φ Definition Let [H] be H with no labels. E φ [ th,φ (G) ] = t [H] (G) Hamed Hatami (McGill University) December 4, / 43
46 Recall that: t H1 H 2 (G) = t H1 (G)t H2 (G). H1 G H2 Hamed Hatami (McGill University) December 4, / 43
47 Recall that: t H1 H 2 (G) = t H1 (G)t H2 (G). H1 G H2 This motivates us to define H 1 H 2 := H 1 H 2. Hamed Hatami (McGill University) December 4, / 43
48 Definition The product H 1 H 2 of partially labeled graphs H 1 and H 2 : Hamed Hatami (McGill University) December 4, / 43
49 Definition The product H 1 H 2 of partially labeled graphs H 1 and H 2 : Take their disjoint union, and then identify vertices with the same label. Hamed Hatami (McGill University) December 4, / 43
50 Definition The product H 1 H 2 of partially labeled graphs H 1 and H 2 : Take their disjoint union, and then identify vertices with the same label. If multiple edges arise, only one copy is kept. Example = Hamed Hatami (McGill University) December 4, / 43
51 Let H 1 and H 2 be partially labeled with labels L 1 and L 2. Hamed Hatami (McGill University) December 4, / 43
52 Let H 1 and H 2 be partially labeled with labels L 1 and L 2. Let φ : L 1 L 2 G. Hamed Hatami (McGill University) December 4, / 43
53 Let H 1 and H 2 be partially labeled with labels L 1 and L 2. Let φ : L 1 L 2 G. We have t H1,φ(G)t H2,φ(G) = t H1 H 2,φ(G). Example G G φ = H1 H2 H1 H2 Hamed Hatami (McGill University) December 4, / 43
54 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Hamed Hatami (McGill University) December 4, / 43
55 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Hamed Hatami (McGill University) December 4, / 43
56 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Let b 1,..., b k be real numbers and φ : L G. Hamed Hatami (McGill University) December 4, / 43
57 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Let b 1,..., b k be real numbers and φ : L G. 0 ( ) 2 bi t Hi,φ(G) = bi b j t Hi,φ(G)t Hj,φ(G) = b i b j t Hi H j,φ(g) Hamed Hatami (McGill University) December 4, / 43
58 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Let b 1,..., b k be real numbers and φ : L G. bi b j t Hi H j,φ(g) 0 Hamed Hatami (McGill University) December 4, / 43
59 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Let b 1,..., b k be real numbers and φ : L G. bi b j t Hi H j,φ(g) 0 [ ] E φ bi b j t Hi H j,φ(g) 0 Hamed Hatami (McGill University) December 4, / 43
60 We want to understand the set of all valid inequalities of the form: For all G a 1 t H1 (G) a k t Hk (G) 0. Let H 1,..., H k be partially labeled graphs with the set of labels L. Let b 1,..., b k be real numbers and φ : L G. bi b j t Hi H j,φ(g) 0 [ ] E φ bi b j t Hi H j,φ(g) 0 bi b j t [Hi H j] (G) 0 Hamed Hatami (McGill University) December 4, / 43
61 Reflection positivity bi b j t [Hi H j] (G) 0 Hamed Hatami (McGill University) December 4, / 43
62 Reflection positivity bi b j t [Hi H j] (G) 0 Let H 1, H 2,... be all partially labeled graphs. For every G: Hamed Hatami (McGill University) December 4, / 43
63 Reflection positivity bi b j t [Hi H j] (G) 0 Let H 1, H 2,... be all partially labeled graphs. For every G: Condition I: t K1 (G) = 1. Hamed Hatami (McGill University) December 4, / 43
64 Reflection positivity bi b j t [Hi H j] (G) 0 Let H 1, H 2,... be all partially labeled graphs. For every G: Condition I: t K1 (G) = 1. Condition II: t H K1 (G) = t H (G) for all graph H. Hamed Hatami (McGill University) December 4, / 43
65 Reflection positivity bi b j t [Hi H j] (G) 0 Let H 1, H 2,... be all partially labeled graphs. For every G: Condition I: t K1 (G) = 1. Condition II: t H K1 (G) = t H (G) for all graph H. Condition III: The infinite matrix whose ij-th entry is t [Hi H j] (G) is positive semi-definite. Hamed Hatami (McGill University) December 4, / 43
66 Reflection positivity bi b j t [Hi H j] (G) 0 Let H 1, H 2,... be all partially labeled graphs. For every G: Condition I: t K1 (G) = 1. Condition II: t H K1 (G) = t H (G) for all graph H. Condition III: The infinite matrix whose ij-th entry is t [Hi H j] (G) is positive semi-definite. Theorem (Freedman, Lovász, Shrijver 2007) These conditions describe the closure of the set {(t F1 (G), t F2 (G),...) : G} [0, 1] N. Hamed Hatami (McGill University) December 4, / 43
67 Quantum Graphs Hamed Hatami (McGill University) December 4, / 43
68 A quantum graph is a formal linear combination of graphs: a 1 H a k H k. Hamed Hatami (McGill University) December 4, / 43
69 A quantum graph is a formal linear combination of graphs: a 1 H a k H k. A quantum graph a 1 H a k H k is called positive, if for all G, a 1 t H1 (G) a k t Hk (G) 0. Hamed Hatami (McGill University) December 4, / 43
70 A quantum graph is a formal linear combination of graphs: a 1 H a k H k. A quantum graph a 1 H a k H k is called positive, if for all G, a 1 t H1 (G) a k t Hk (G) 0. Goodman: K 3 2(K 2 K 2 ) + K 2 0. Hamed Hatami (McGill University) December 4, / 43
71 A quantum graph is a formal linear combination of graphs: a 1 H a k H k. A quantum graph a 1 H a k H k is called positive, if for all G, a 1 t H1 (G) a k t Hk (G) 0. Goodman: K 3 2(K 2 K 2 ) + K 2 0. We want to understand the set of all positive quantum graphs. Hamed Hatami (McGill University) December 4, / 43
72 A partially labeled quantum graph is a formal linear combination of partially labeled graphs: a 1 H a k H k. Hamed Hatami (McGill University) December 4, / 43
73 A partially labeled quantum graph is a formal linear combination of partially labeled graphs: a 1 H a k H k. Partially labeled quantum graphs form an algebra: (a 1 H a k H k ) (b 1 L b l L l ) = a i b j H i L j. Hamed Hatami (McGill University) December 4, / 43
74 Unlabeling operator [ ] : partially labeled quantum graph quantum graph Hamed Hatami (McGill University) December 4, / 43
75 Unlabeling operator [ ] : partially labeled quantum graph quantum graph Recall [ ( b i H i ) 2] = b i b j [ Hi H j ] 0 Hamed Hatami (McGill University) December 4, / 43
76 Unlabeling operator [ ] : partially labeled quantum graph quantum graph Recall [ ( b i H i ) 2] = b i b j [ Hi H j ] 0 Equivalently For every partially labeled quantum graph g we have [ g 2] 0. Hamed Hatami (McGill University) December 4, / 43
77 Unlabeling operator [ ] : partially labeled quantum graph quantum graph Recall [ ( b i H i ) 2] = b i b j [ Hi H j ] 0 Equivalently For every partially labeled quantum graph g we have [ g 2] 0. Corollary Always [ ] g g2 k 0. Hamed Hatami (McGill University) December 4, / 43
78 Question (Lovász s 17th Problem, Lovász-Szegedy, Razborov) Is it true that every f 0 is of the form [ ] f = g1 2 + g g2 k Hamed Hatami (McGill University) December 4, / 43
79 Question (Lovász s 17th Problem, Lovász-Szegedy, Razborov) Is it true that every f 0 is of the form [ ] f = g1 2 + g g2 k Observation (Lovasz-Szegedy and Razborov) If f 0 and ɛ > 0, there exists a positive integer k and quantum labeled graphs g 1, g 2,..., g k such that [ ] ɛ f g1 2 + g g2 k ɛ. Hamed Hatami (McGill University) December 4, / 43
80 Question (Lovász s 17th Problem, Lovász-Szegedy, Razborov) Is it true that every f 0 is of the form [ ] f = g1 2 + g g2 k Observation (Lovasz-Szegedy and Razborov) If f 0 and ɛ > 0, there exists a positive integer k and quantum labeled graphs g 1, g 2,..., g k such that [ ] ɛ f g1 2 + g g2 k ɛ. Theorem (HH and Norin) The answer to the above question is negative. Hamed Hatami (McGill University) December 4, / 43
81 positive polynomials Hamed Hatami (McGill University) December 4, / 43
82 Polynomial p R[x 1,..., x n ] is called positive if it takes only non-negative values. Hamed Hatami (McGill University) December 4, / 43
83 Polynomial p R[x 1,..., x n ] is called positive if it takes only non-negative values. p p2 k is always positive. Hamed Hatami (McGill University) December 4, / 43
84 Polynomial p R[x 1,..., x n ] is called positive if it takes only non-negative values. p p2 k is always positive. Theorem (Hilbert 1888) There exist 3-variable positive homogenous polynomials which are not sums of squares of polynomials. Hamed Hatami (McGill University) December 4, / 43
85 Polynomial p R[x 1,..., x n ] is called positive if it takes only non-negative values. p p2 k is always positive. Theorem (Hilbert 1888) There exist 3-variable positive homogenous polynomials which are not sums of squares of polynomials. Example (Motzkin s polynomial) x 4 y 2 + y 4 z 2 + z 4 x 2 6x 2 y 2 z 2 0. Hamed Hatami (McGill University) December 4, / 43
86 Extending to quantum graphs Hamed Hatami (McGill University) December 4, / 43
87 Theorem (HH and Norin) There are positive quantum graphs f which are not sums of squares. That is, always f [ g g2 k ]. Hamed Hatami (McGill University) December 4, / 43
88 Theorem (HH and Norin) There are positive quantum graphs f which are not sums of squares. That is, always f [ g g2 k ]. The proof is based on converting x 4 y 2 + y 4 z 2 + z 4 x 2 6x 2 y 2 z 2 to a quantum graph. Hamed Hatami (McGill University) December 4, / 43
89 Theorem (Artin 1927, Solution to Hilbert s 17th Problem) Every positive polynomial is of the form (p 1 /q 1 ) (p k /q k ) 2. Hamed Hatami (McGill University) December 4, / 43
90 Theorem (Artin 1927, Solution to Hilbert s 17th Problem) Every positive polynomial is of the form (p 1 /q 1 ) (p k /q k ) 2. Corollary The problem of checking the positivity of a polynomial is decidable. Hamed Hatami (McGill University) December 4, / 43
91 Theorem (Artin 1927, Solution to Hilbert s 17th Problem) Every positive polynomial is of the form Corollary (p 1 /q 1 ) (p k /q k ) 2. The problem of checking the positivity of a polynomial is decidable. Co-recursively enumerable: Try to find a point that makes p negative. Hamed Hatami (McGill University) December 4, / 43
92 Theorem (Artin 1927, Solution to Hilbert s 17th Problem) Every positive polynomial is of the form (p 1 /q 1 ) (p k /q k ) 2. Corollary The problem of checking the positivity of a polynomial is decidable. Co-recursively enumerable: Try to find a point that makes p negative. recursively enumerable: Try to write p = (p i /q i ) 2. Hamed Hatami (McGill University) December 4, / 43
93 Our solution to Lovász s 17th problem was based on an analogy to polynomials. Hamed Hatami (McGill University) December 4, / 43
94 Our solution to Lovász s 17th problem was based on an analogy to polynomials. Since there are polynomials which are positive but not sums of squares, our theorem was expected. Hamed Hatami (McGill University) December 4, / 43
95 Our solution to Lovász s 17th problem was based on an analogy to polynomials. Since there are polynomials which are positive but not sums of squares, our theorem was expected. Lovász: Does Artin s theorem (sums of rational functions) hold for graph homomorphisms? Hamed Hatami (McGill University) December 4, / 43
96 Our solution to Lovász s 17th problem was based on an analogy to polynomials. Since there are polynomials which are positive but not sums of squares, our theorem was expected. Lovász: Does Artin s theorem (sums of rational functions) hold for graph homomorphisms? Maybe at least the decidability? (A 10th problem) Hamed Hatami (McGill University) December 4, / 43
97 Our solution to Lovász s 17th problem was based on an analogy to polynomials. Since there are polynomials which are positive but not sums of squares, our theorem was expected. Lovász: Does Artin s theorem (sums of rational functions) hold for graph homomorphisms? Maybe at least the decidability? (A 10th problem) Theorem (HH and Norin) The following problem is undecidable. QUESTION: Does the inequality a 1 t H1 (G) a k t Hk (G) 0 hold for every graph G? Hamed Hatami (McGill University) December 4, / 43
98 Proof Hamed Hatami (McGill University) December 4, / 43
99 Theorem (HH and Norin) The following problem is undecidable. QUESTION: Does the inequality a 1 t H1 (G) a k t Hk (G) 0 hold for every graph G? Hamed Hatami (McGill University) December 4, / 43
100 Theorem (HH and Norin) The following problem is undecidable. QUESTION: Does the inequality a 1 t H1 (G) a k t Hk (G) 0 hold for every graph G? Equivalently Theorem (HH and Norin) The following problem is undecidable. INSTANCE: A polynomial p(x 1,..., x k ) and graphs H 1,..., H k. QUESTION: Does the inequality p(t H1 (G),..., t Hk (G)) 0 hold for every graph G? Hamed Hatami (McGill University) December 4, / 43
101 Theorem (HH and Norin) The following problem is undecidable. INSTANCE: A polynomial p(x 1,..., x k ) and graphs H 1,..., H k. QUESTION: Does the inequality p(t H1 (G),..., t Hk (G)) 0 hold for every graph G? Hamed Hatami (McGill University) December 4, / 43
102 Theorem (HH and Norin) The following problem is undecidable. INSTANCE: A polynomial p(x 1,..., x k ) and graphs H 1,..., H k. QUESTION: Does the inequality p(t H1 (G),..., t Hk (G)) 0 hold for every graph G? Instead I will prove the following theorem: Theorem The following problem is undecidable. INSTANCE: A polynomial p(x 1,..., x k, y 1,..., y k ). QUESTION: Does the inequality p(t K2 (G 1 ),..., t K2 (G k ), t K3 (G 1 ),..., t K3 (G k )) 0 hold for every G 1,..., G k? Hamed Hatami (McGill University) December 4, / 43
103 Matiyasevich 1970 Solution to Hilbert s 10th problem: Checking the positivity of p R[x 1,..., x k ] on { 1 1 n : n Z} k is undecidable. Hamed Hatami (McGill University) December 4, / 43
104 Matiyasevich 1970 Solution to Hilbert s 10th problem: Checking the positivity of p R[x 1,..., x k ] on { 1 1 n : n Z} k is undecidable. Bollobás, Razborov: Goodman s bound is achieved only when t K2 (G) { 1 1 n : n Z}. 1 t(k 3 ; G) 1/2 Kruskal-Katona Goodman 4/5 3/4 2/3 Razborov 1/2 t(k 2 ; G) 1 Hamed Hatami (McGill University) December 4, / 43
105 Let S be the grey area and g(x) = 2x 2 x. (Goodman: t K3 (G) 2t K2 (G) 2 t K2 (G).) 1 t(k 3 ; G) 1/2 Kruskal-Katona Goodman 4/5 3/4 2/3 Razborov 1/2 t(k 2 ; G) 1 Hamed Hatami (McGill University) December 4, / 43
106 Let S be the grey area and g(x) = 2x 2 x. (Goodman: t K3 (G) 2t K2 (G) 2 t K2 (G).) Lemma Let p R[x 1,..., x k ]. Define q(x 1,..., x k, y 1,..., y k ) as T.F.A.E. q := p ( k k ) (1 x i ) 6 + C p y i g(x i ). i=1 i=1 Hamed Hatami (McGill University) December 4, / 43
107 Let S be the grey area and g(x) = 2x 2 x. (Goodman: t K3 (G) 2t K2 (G) 2 t K2 (G).) Lemma Let p R[x 1,..., x k ]. Define q(x 1,..., x k, y 1,..., y k ) as T.F.A.E. q := p ( k k ) (1 x i ) 6 + C p y i g(x i ). i=1 p < 0 for some x 1,..., x k {1 1/n : n N}. (undecidable) i=1 Hamed Hatami (McGill University) December 4, / 43
108 Let S be the grey area and g(x) = 2x 2 x. (Goodman: t K3 (G) 2t K2 (G) 2 t K2 (G).) Lemma Let p R[x 1,..., x k ]. Define q(x 1,..., x k, y 1,..., y k ) as T.F.A.E. q := p ( k k ) (1 x i ) 6 + C p y i g(x i ). i=1 p < 0 for some x 1,..., x k {1 1/n : n N}. (undecidable) q < 0 for some (x i, y i ) S s. i=1 Hamed Hatami (McGill University) December 4, / 43
109 Let S be the grey area and g(x) = 2x 2 x. (Goodman: t K3 (G) 2t K2 (G) 2 t K2 (G).) Lemma Let p R[x 1,..., x k ]. Define q(x 1,..., x k, y 1,..., y k ) as T.F.A.E. q := p ( k k ) (1 x i ) 6 + C p y i g(x i ). i=1 p < 0 for some x 1,..., x k {1 1/n : n N}. (undecidable) q < 0 for some (x i, y i ) S s. q < 0 for some x i = t K2 (G i ) and y i = t K3 (G i ). (reduction) i=1 Hamed Hatami (McGill University) December 4, / 43
110 Where do we go from here? Hamed Hatami (McGill University) December 4, / 43
111 On can hope decidability for restricted classes of graphs. Hamed Hatami (McGill University) December 4, / 43
112 On can hope decidability for restricted classes of graphs. Bollobas: Linear inequalities a 1 K n a k K nk 0 is decidable. Hamed Hatami (McGill University) December 4, / 43
113 On can hope decidability for restricted classes of graphs. Bollobas: Linear inequalities a 1 K n a k K nk 0 is decidable. Question: What about unions of cliques? Hamed Hatami (McGill University) December 4, / 43
114 Let R denote the closure of {(t H1 (G), t H2 (G),...) : G} [0, 1] N. R Hamed Hatami (McGill University) December 4, / 43
115 Graphons: The points in R (graph limits) can be represented by symmetric measurable W : [0, 1] 2 [0, 1]. Hamed Hatami (McGill University) December 4, / 43
116 Graphons: The points in R (graph limits) can be represented by symmetric measurable W : [0, 1] 2 [0, 1]. Finitely Forcible: A point in R is finitely forcible if a finite number of coordinates uniquely determine it. Hamed Hatami (McGill University) December 4, / 43
117 Graphons: The points in R (graph limits) can be represented by symmetric measurable W : [0, 1] 2 [0, 1]. Finitely Forcible: A point in R is finitely forcible if a finite number of coordinates uniquely determine it. Lovász s Conjecture: Every feasible inequality a 1 t H1 (W ) a k t Hk (W ) < 0 has a finitely forcible solution W. Hamed Hatami (McGill University) December 4, / 43
118 Graphons: The points in R (graph limits) can be represented by symmetric measurable W : [0, 1] 2 [0, 1]. Finitely Forcible: A point in R is finitely forcible if a finite number of coordinates uniquely determine it. Lovász s Conjecture: Every feasible inequality a 1 t H1 (W ) a k t Hk (W ) < 0 has a finitely forcible solution W. Lovász-Szegedy s Conjecture: Finitely forcible graphons have simple structures (finite dimensional). Hamed Hatami (McGill University) December 4, / 43
119 Graphons: The points in R (graph limits) can be represented by symmetric measurable W : [0, 1] 2 [0, 1]. Finitely Forcible: A point in R is finitely forcible if a finite number of coordinates uniquely determine it. Lovász s Conjecture: Every feasible inequality a 1 t H1 (W ) a k t Hk (W ) < 0 has a finitely forcible solution W. Lovász-Szegedy s Conjecture: Finitely forcible graphons have simple structures (finite dimensional). [Glebov, Klimošová, Král 2013+] There are finitely forcible W s such that {W (x, ) : x [0, 1]} with the L 1 distance contains a subset homeomorphic to [0, 1]. Hamed Hatami (McGill University) December 4, / 43
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