Efficient arithmetic regularity and removal lemmas for induced bipartite patterns Yufei Zhao (MIT) Joint work with Noga Alon (Princeton) and Jacob Fox (Stanford) April 22, 2018 1
Szemer´ edi’s graph regularity lemma Graph regularity lemma For every ǫ > 0 there exists M = M ( ǫ ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular 2
Szemer´ edi’s graph regularity lemma Graph regularity lemma For every ǫ > 0 there exists M = M ( ǫ ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular Graph removal lemma For every ∀ ǫ > 0 and graph H there is some δ = δ ( H , ǫ ) > 0 so that every n -vertex graph with H -density < δ can be made H -free by removing < ǫ n 2 edges 2
Szemer´ edi’s graph regularity lemma Graph regularity lemma For every ǫ > 0 there exists M = M ( ǫ ) so that every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular Graph removal lemma For every ∀ ǫ > 0 and graph H there is some δ = δ ( H , ǫ ) > 0 so that every n -vertex graph with H -density < δ can be made H -free by removing < ǫ n 2 edges ◮ M ( ǫ ) = 2 2 2 ... 2 tower of height ǫ − O (1) (cannot be improved [Gowers]) ◮ Removal lemma holds with δ = M − O (1) = 1 / 2 2 2 ... 2 (possibly could be improved, but not beyond ǫ C log(1 /ǫ ) when H = K 3 ) 2
When can you guarantee poly(1 /ǫ ) bounds? 3
When can you guarantee poly(1 /ǫ ) bounds? For a graph with bounded VC dimension: ◮ Vertices can be partitioned into ǫ − O (1) parts ◮ All but ǫ -fraction of pairs of vertex parts have densities ≤ ǫ or ≥ 1 − ǫ [Alon–Fischer–Newman, Lov´ asz–Szegedy] 3
What is VC dimension? Let S be a collection of subsets of Ω dim VC S := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U ′ ⊆ U there exists T ∈ S such that T ∩ U = U ′ 4
What is VC dimension? Let S be a collection of subsets of Ω dim VC S := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U ′ ⊆ U there exists T ∈ S such that T ∩ U = U ′ E.g., the VC-dimension of the collection of half-planes in R 2 is 3 4
What is VC dimension? Let S be a collection of subsets of Ω dim VC S := size of the largest shattered subset of Ω U ⊂ Ω is shattered if for every U ′ ⊆ U there exists T ∈ S such that T ∩ U = U ′ E.g., the VC-dimension of the collection of half-planes in R 2 is 3 VC dimension of a graph G is defined to be the VC dimension of the collection of vertex neighborhoods (Ω = V ( G )): dim VC G := dim VC { N ( v ) : v ∈ V ( G ) } 4
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph 5
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) 5
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) 5
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care 5
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care dim VC G < d ⇐ ⇒ G forbids the following as a bi-induced subgraph: 1 00 1 000 0 1 001 1 01 2 010 011 10 3 2 100 101 11 110 111 d = 1 d = 2 d = 3 5
Bounded VC dimension ⇐ ⇒ forbidding a bi-induced subgraph H G H as a subgraph of G (all edges of H are present in G ) H as an induced subgraph of G (all edges of H are present H G in G and non-edges are not present) don’t care Bipartite H as a bi-induced subgraph (similar to induced H G but don’t care about edges inside each bipartition) don’t care dim VC G < d ⇐ ⇒ G forbids the following as a bi-induced subgraph: 1 00 1 000 0 1 001 1 01 2 010 011 10 3 2 100 101 11 110 111 d = 1 d = 2 d = 3 Conversely, if G is bi-induced- H -free, then dim VC G = O H (1) 5
When can you guarantee poly(1 /ǫ ) bounds? Hereditary family – any family of graphs closed under deletion of vertices. ◮ E.g., 3-colorable, planar, bipartite, triangle-free, chordal, perfect ◮ Equivalent to being induced- H -free for all H in some (possibly infinite) family H 6
When can you guarantee poly(1 /ǫ ) bounds? Hereditary family – any family of graphs closed under deletion of vertices. ◮ E.g., 3-colorable, planar, bipartite, triangle-free, chordal, perfect ◮ Equivalent to being induced- H -free for all H in some (possibly infinite) family H For any given hereditary family F for graphs: ◮ If graphs in F have bounded VC-dimension, then every graph has an ǫ -regular partition into ǫ − O (1) parts. 6
When can you guarantee poly(1 /ǫ ) bounds? Hereditary family – any family of graphs closed under deletion of vertices. ◮ E.g., 3-colorable, planar, bipartite, triangle-free, chordal, perfect ◮ Equivalent to being induced- H -free for all H in some (possibly infinite) family H For any given hereditary family F for graphs: ◮ If graphs in F have bounded VC-dimension, then every graph has an ǫ -regular partition into ǫ − O (1) parts. ◮ If graphs in F do not have bounded VC-dimension, then there exist graphs in F whose ǫ -regular partition whose number of parts is necessarily at least 2 2 ... 2 (tower height ǫ − c ) 6
When can you guarantee poly (1 /ǫ ) bounds? Regularity lemma for graphs of bounded VC dimension For a fixed bipartite H , if G is bi-induced- H -free, then G has a vertex partition into ǫ − O (1) parts so that all but ≤ ǫ -fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ . 7
When can you guarantee poly (1 /ǫ ) bounds? Regularity lemma for graphs of bounded VC dimension For a fixed bipartite H , if G is bi-induced- H -free, then G has a vertex partition into ǫ − O (1) parts so that all but ≤ ǫ -fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ . A graph is k -stable if it does not contain a bi-induced half-graph on 2 k vertices. 7
When can you guarantee poly (1 /ǫ ) bounds? Regularity lemma for graphs of bounded VC dimension For a fixed bipartite H , if G is bi-induced- H -free, then G has a vertex partition into ǫ − O (1) parts so that all but ≤ ǫ -fraction of pairs have edge-densities ≤ ǫ or ≥ 1 − ǫ . A graph is k -stable if it does not contain a bi-induced half-graph on 2 k vertices. Stable regularity lemma [Malliaris–Shelah] If the graph is k -stable, then we can furthermore guarantee that every pair of parts has density ≤ ǫ or ≥ 1 − ǫ . 7
Arithmetic setting G abelian group, A ⊂ G dim VC A := dim VC { A + x : x ∈ G } = dim VC CayleyGraph ( G , A ) 8
Arithmetic setting G abelian group, A ⊂ G dim VC A := dim VC { A + x : x ∈ G } = dim VC CayleyGraph ( G , A ) We say that A contains a bi-induced copy of a bipartite graph H if the same is true for CayleyGraph ( G , A ) 8
Arithmetic regularity lemma ◮ Szemer´ edi’s graph regularity lemma : ∀ ǫ > 0 ∃ M : every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular 9
Arithmetic regularity lemma ◮ Szemer´ edi’s graph regularity lemma : ∀ ǫ > 0 ∃ M : every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular ◮ Arithmetic regularity lemma [Green]: ∀ ǫ > 0 , p ∃ M : for every A ⊂ F n p , there is some V ≤ F n 2 of codimension ≤ M such that all but ≤ ǫ fraction of translates of V meet A in an ǫ -Fourier uniform way 9
Arithmetic regularity lemma ◮ Szemer´ edi’s graph regularity lemma : ∀ ǫ > 0 ∃ M : every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular ◮ Arithmetic regularity lemma [Green]: ∀ ǫ > 0 , p ∃ M : for every A ⊂ F n p , there is some V ≤ F n 2 of codimension ≤ M such that all but ≤ ǫ fraction of translates of V meet A in an ǫ -Fourier uniform way ◮ Corollary: removal lemma for arithmetic patterns 9
Arithmetic regularity lemma ◮ Szemer´ edi’s graph regularity lemma : ∀ ǫ > 0 ∃ M : every graph has a vertex partition into ≤ M parts so that all but < ǫ fraction of pairs are ǫ -regular ◮ Arithmetic regularity lemma [Green]: ∀ ǫ > 0 , p ∃ M : for every A ⊂ F n p , there is some V ≤ F n 2 of codimension ≤ M such that all but ≤ ǫ fraction of translates of V meet A in an ǫ -Fourier uniform way ◮ Corollary: removal lemma for arithmetic patterns (later shown to follow from graph removal lemma [Kr´ al’–Serra–Vena / Shapira]) 9
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