Background and previous results Our results Summary Multicolor Hypergraph Containers Victor Falgas-Ravry 1 Kelly O’Connell 2 Johanna Strömberg 3 Andrew J. Uzzell 4 1 Umeå University 2 Vanderbilt University 3 Uppsala University 4 University of Nebraska–Lincoln May 21, 2016
Background and previous results Our results Summary Overview Today, I will talk about: the hypergraph container method and its applications; hereditary properties of graphs and multicolored graphs (including oriented and directed graphs); how to use hypergraph containers to determine the number of multicolored graphs that satisfy a given hereditary property.
Background and previous results Our results Summary Outline Background and previous results 1 Hypergraph containers and their applications Hereditary properties of graphs Our results 2 Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications
Background and previous results Our results Summary Outline Background and previous results 1 Hypergraph containers and their applications Hereditary properties of graphs Our results 2 Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications
Background and previous results Our results Summary Outline Background and previous results 1 Hypergraph containers and their applications Hereditary properties of graphs Our results 2 Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications
Background and previous results Our results Summary Definitions and notation: H -free (hyper)graphs Definition Let H be a graph. We say that a graph G is H -free if H is not a subgraph of G . (If H is a hypergraph, the definition of H -free is the same.) Definition Let H be a graph. The Turán number of H , denoted ex ( n , H ) , is the maximum number of edges in an H -free graph on n vertices.
Background and previous results Our results Summary Definitions and notation: hypergraphs and independent sets Definitions Let H be a hypergraph. We say that I ⊆ V ( H ) is an independent set if no edge of H is contained entirely in I . α ( H ) denotes the maximum size of an independent set in H . I ( H ) denotes the family of independent sets in H i ( H ) := |I ( H ) | denotes the number of independent sets in H . Definition An r -uniform hypergraph (or r -graph ) is a hypergraph in which each edge contains exactly r vertices.
Background and previous results Our results Summary The hypergraph G ( n , K 3 ) Definition Given n , let G ( n , K 3 ) be the 3-graph with vertex set E ( K n ) such that { e 1 , e 2 , e 3 } ∈ E ( G ( n , K 3 )) if and only if { e 1 , e 2 , e 3 } , considered as a graph, forms a copy of K 3 . a f d a d b f e c b c e
Background and previous results Our results Summary Two well-known theorems concerning G ( n , K 3 ) Definition Given n , let G ( n , K 3 ) be the 3-graph with vertex set E ( K n ) such that { e 1 , e 2 , e 3 } ∈ E ( G ( n , K 3 )) if and only if { e 1 , e 2 , e 3 } , considered as a graph, forms a copy of K 3 . Theorem ( ) � n 2 � � � G ( n , K 3 ) = . α 4 Theorem ( ) � � n 2 4 + o ( n 2 ) . G ( n , K 3 ) = 2 i
Background and previous results Our results Summary Two well-known theorems concerning G ( n , K 3 ) Definition Given n , let G ( n , K 3 ) be the 3-graph with vertex set E ( K n ) such that { e 1 , e 2 , e 3 } ∈ E ( G ( n , K 3 )) if and only if { e 1 , e 2 , e 3 } , considered as a graph, forms a copy of K 3 . Theorem (Mantel, 1907) � n 2 � � � G ( n , K 3 ) = . α 4 Theorem (Erdős, Kleitman, & Rothschild, 1976) � � n 2 4 + o ( n 2 ) . G ( n , K 3 ) = 2 i
Background and previous results Our results Summary Well-known theorems reinterpreted Observations There is a 1-to-1 correspondence between triangle-free graphs on [ n ] and independent sets in G ( n , K 3 ) . In particular, ex ( n , K 3 ) = α ( G ( n , K 3 )) and the number of triangle-free graphs on [ n ] equals i ( G ( n , K 3 )) . a f d a d b f e c b c e
Background and previous results Our results Summary Problems involving independent sets in hypergraphs We have translated extremal problems into questions about independent sets in hypergraphs. Problem For many hypergraphs H , I ( H ) is a large, complicated family. Solution Replace I ( H ) with a smaller, simpler family.
Background and previous results Our results Summary Hypergraph containers What we want Given a hypergraph H of order N and ε > 0, we want a collection C of subsets of V ( H ) such that: 1 C is a container family for I ( H ) . 2 C is small . 3 Elements of C are almost independent .
Background and previous results Our results Summary Hypergraph containers What we want Given a hypergraph H of order N and ε > 0, we want a collection C of subsets of V ( H ) such that: 1 C is a container family for I ( H ) . (Every I ∈ I ( H ) is a subset of some C ∈ C .) 2 C is small . ( |C| ≤ 2 ε N .) 3 Elements of C are almost independent . (For all C ∈ C , e ( H [ C ]) ≤ ε e ( H ) .)
Background and previous results Our results Summary Hypergraph containers What we want Given a hypergraph H of order N and ε > 0, we want a collection C of subsets of V ( H ) such that: 1 C is a container family for I ( H ) . (Every I ∈ I ( H ) is a subset of some C ∈ C .) 2 C is small . ( |C| ≤ 2 ε N .) 3 Elements of C are almost independent . (For all C ∈ C , e ( H [ C ]) ≤ ε e ( H ) .) Balogh, Morris, & Samotij (2015) and Saxton & Thomason (2015) independently showed that every hypergraph has such a container family. Graph containers were first used by Kleitman & Winston (1982) and by Sapozhenko (1987–2003).
Background and previous results Our results Summary A hypergraph container family for G ( n , K 3 ) We count triangle-free graphs by finding a container family for G ( n , K 3 ) . Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [ n ] such that 1 Every triangle-free graph on [ n ] is a subgraph of some C ∈ C . 2 |C| ≤ 2 ε n 2 . 3 If C ∈ C , then C contains at most ε n 3 triangles and � 1 �� n � e ( C ) ≤ 2 + ε . 2
Background and previous results Our results Summary A hypergraph container family for G ( n , K 3 ) We count triangle-free graphs by finding a container family for G ( n , K 3 ) . Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [ n ] such that 1 Every triangle-free graph on [ n ] is a subgraph of some C ∈ C . 2 |C| ≤ 2 ε n 2 . 3 If C ∈ C , then C contains at most ε n 3 triangles and � 1 �� n � e ( C ) ≤ 2 + ε . 2 Remark: By a “supersaturation” result of Erdős and Simonovits, � 1 �� n � if a graph C has at most ε n 3 triangles, then e ( C ) ≤ 2 + ε . 2 Applications of the container method nearly always rely on supersaturation results.
Background and previous results Our results Summary Using hypergraph containers to count triangle-free graphs Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [ n ] such that 1 Every triangle-free graph on [ n ] is a subgraph of some C ∈ C . 2 |C| ≤ 2 ε n 2 . 3 If C ∈ C , then C contains at most ε n 3 triangles and � 1 �� n � e ( C ) ≤ 2 + ε . 2 Let P n denote the class of triangle-free graphs on [ n ] . Every C ∈ C has at most 2 e ( C ) triangle-free subgraphs. The theorem then implies that 4 + ε n 2 = 2 n 2 n 2 |P n | ≤ |C|· 2 max C ∈ C e ( C ) ≤ 2 ε n 2 2 4 + o ( n 2 ) . For the lower bound, take all 2 n 2 / 4 subgraphs of K n / 2 , n / 2 .
Background and previous results Our results Summary Applications of hypergraph containers: counting (Hyper)graph containers have been used to count a variety of discrete objects: H -free (hyper)graphs of order n (Kleitman & Winston (1982); Balogh & Samotij (2011, 2011); Balogh, Morris, & Samotij (2015); Saxton & Thomason (2015); Morris & Saxton (2016+)) Antichains in the Boolean lattice (Balogh, Treglown, & Wagner (2016+)) Discrete metric spaces with a specified number of distances (Balogh & Wagner (2016)) t -error-correcting binary codes (Balogh, Treglown, & Wagner (2016+)) Sum-free subsets of [ n ] and of abelian groups (Sapozhenko (2003); Alon, Balogh, Morris, & Samotij (2014)) Matroids on n elements (Bansal, Pendavingh, & van der Pol (2015))
Background and previous results Our results Summary Applications of hypergraph containers: typical structure Containers can also be used to characterize the typical structure of discrete objects: H -free (hyper)graphs (Balogh, Morris, & Samotij (2015)) Edge-maximal triangle-free graphs (Balogh, Liu, Petříčková, & Sharifzadeh (2015)) Graphs without cliques of order r , where r grows slowly with n (Balogh, Bushaw, Collares Neto, Liu, Morris, & Sharifzadeh (2016+)) Graphs without induced even cycles (Kim, Kühn, Osthus, & Townsend (2015+))
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