The Rainbow Tur´ an Problem for Even Cycles Shagnik Das University of California, Los Angeles Aug 20, 2012 Joint work with Choongbum Lee and Benny Sudakov
Historical Background Rainbow Tur´ an Problem Our Results Plan Historical Background 1 Tur´ an Problems Colouring Problems Rainbow Tur´ an Problem 2 Definition Motivation Known Results Our Results 3 Summary Warm Up Sketch of Proof
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem Mantel’s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n 2 4 edges.
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem Mantel’s theorem: most fundamental in extremal graph theory Theorem (Mantel, 1907) If a graph G on n vertices has no triangle, then G has at most n 2 4 edges. Tur´ an generalised to cliques of any order Theorem (Tur´ an, 1941) If a graph G on n vertices has no clique of order r, then G has at � � n 2 1 most 1 − r − 1 + o (1) 2 edges.
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Extended Can define Tur´ an numbers of general graphs Definition (Tur´ an numbers) Given any graph H , we define the Tur´ an number ex ( n , H ) to be the maximum number of edges in an H -free graph on n vertices.
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Extended Can define Tur´ an numbers of general graphs Definition (Tur´ an numbers) Given any graph H , we define the Tur´ an number ex ( n , H ) to be the maximum number of edges in an H -free graph on n vertices. Erd˝ os and Stone found asymptotics for all non-bipartite graphs Theorem (Erd˝ os-Stone, 1946) = 1 − ( χ ( H ) − 1) − 1 . � n � For all graphs H, lim n →∞ ex ( n , H ) / 2
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Open Problems Tur´ an problem for bipartite graphs generally open
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Open Problems Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Open Problems Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) � n 1+ 1 � For all k ≥ 2 , ex ( n , C 2 k ) = O . k
Historical Background Rainbow Tur´ an Problem Our Results Tur´ an’s Theorem: Open Problems Tur´ an problem for bipartite graphs generally open Particularly interesting is the case of even cycles Conjectured upper bound known Theorem (Bondy-Simonovits, 1974) � n 1+ 1 � For all k ≥ 2 , ex ( n , C 2 k ) = O . k Matching lower bound only known for k = 2 , 3 , 5
Historical Background Rainbow Tur´ an Problem Our Results Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k , l ≥ 1 , there exists R ( k , l ) such that any red-blue colouring of K R ( k , l ) contains either a red K k or a blue K l .
Historical Background Rainbow Tur´ an Problem Our Results Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k , l ≥ 1 , there exists R ( k , l ) such that any red-blue colouring of K R ( k , l ) contains either a red K k or a blue K l . Determining the Ramsey numbers R ( k , l ) a widely open problem
Historical Background Rainbow Tur´ an Problem Our Results Colouring Problems: Ramsey Theory Another central result in extremal combinatorics Theorem (Ramsey, 1930) For any integers k , l ≥ 1 , there exists R ( k , l ) such that any red-blue colouring of K R ( k , l ) contains either a red K k or a blue K l . Determining the Ramsey numbers R ( k , l ) a widely open problem Introduction of the probabilistic method
Historical Background Rainbow Tur´ an Problem Our Results Colouring Problems: Extensions Theorem (Erd˝ os-Rado, 1950) For every t, there is an n such that every edge-colouring of K n has a copy of K t with one of the following canonical colourings: 1 2 1 2 1 2 1 2 3 4 3 4 3 4 3 4 constant rainbow minimum maximum
Historical Background Rainbow Tur´ an Problem Our Results Colouring Problems: Extensions Theorem (Erd˝ os-Rado, 1950) For every t, there is an n such that every edge-colouring of K n has a copy of K t with one of the following canonical colourings: 1 2 1 2 1 2 1 2 3 4 3 4 3 4 3 4 constant rainbow minimum maximum In particular, for every t there is an n such that every proper edge-colouring of K n has a rainbow K t .
Historical Background Rainbow Tur´ an Problem Our Results Rainbow Tur´ an Problem First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete
Historical Background Rainbow Tur´ an Problem Our Results Rainbow Tur´ an Problem First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) an number ex ∗ ( n , H ) Given a graph H , we define the rainbow Tur´ to be the maximum number of edges in a properly edge-coloured n -vertex graph with no rainbow copy of H .
Historical Background Rainbow Tur´ an Problem Our Results Rainbow Tur´ an Problem First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) an number ex ∗ ( n , H ) Given a graph H , we define the rainbow Tur´ to be the maximum number of edges in a properly edge-coloured n -vertex graph with no rainbow copy of H . Trivial bound: ex ( n , H ) ≤ ex ∗ ( n , H )
Historical Background Rainbow Tur´ an Problem Our Results Rainbow Tur´ an Problem First introduced by Keevash, Mubayi, Sudakov, Verstra¨ ete Definition (Rainbow Tur´ an Numbers) an number ex ∗ ( n , H ) Given a graph H , we define the rainbow Tur´ to be the maximum number of edges in a properly edge-coloured n -vertex graph with no rainbow copy of H . Trivial bound: ex ( n , H ) ≤ ex ∗ ( n , H ) Reduction to regular Tur´ an problem ⇒ ex ∗ ( n , H ) ≤ ex ( n , H ) + o ( n 2 )
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Motivation Definition ( B k -sets) A subset A of an abelian group G is a B k -set if every g ∈ G has at most one representation of the form g = a 1 + a 2 + . . . + a k , a i ∈ A .
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Motivation Definition ( B k -sets) A subset A of an abelian group G is a B k -set if every g ∈ G has at most one representation of the form g = a 1 + a 2 + . . . + a k , a i ∈ A . Definition ( B ∗ k -sets) A subset A of an abelian group G is a B ∗ k -set if there are no two disjoint k -sets { x 1 , x 2 , . . . , x k } , { y 1 , y 2 , . . . , y k } ⊂ A with the same sum.
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Motivation Definition ( B k -sets) A subset A of an abelian group G is a B k -set if every g ∈ G has at most one representation of the form g = a 1 + a 2 + . . . + a k , a i ∈ A . Definition ( B ∗ k -sets) A subset A of an abelian group G is a B ∗ k -set if there are no two disjoint k -sets { x 1 , x 2 , . . . , x k } , { y 1 , y 2 , . . . , y k } ⊂ A with the same sum. Example ( B ∗ k -sets need not be B k -sets) Let k = 3, G = Z / 6 Z , and A = { 0 , 1 , 2 , 3 , 4 , 5 } . A is not a B 3 -set: 0 + 1 + 4 = 0 + 2 + 3 = 0 + 0 + 5. A is a B ∗ 3 -set: the sum of all six elements is odd.
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B ∗ k -sets → rainbow- C 2 k -free bipartite graphs
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B ∗ k -sets → rainbow- C 2 k -free bipartite graphs Construct bipartite graph on X ∪ Y , where X , Y = G Edge ( x , y ) iff y − x = a ∈ A ⊂ G , colour a
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B ∗ k -sets → rainbow- C 2 k -free bipartite graphs Construct bipartite graph on X ∪ Y , where X , Y = G Edge ( x , y ) iff y − x = a ∈ A ⊂ G , colour a x 1 y 1 x 2 y 2 y 3 x 3 x 4 y 4 x 5 y 5 Example with G = Z / 5 Z , A = { 1 , 3 }
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B ∗ k -sets → rainbow- C 2 k -free bipartite graphs Construct bipartite graph on X ∪ Y , where X , Y = G Edge ( x , y ) iff y − x = a ∈ A ⊂ G , colour a x 1 y 1 x 2 y 2 y 3 x 3 x 4 y 4 x 5 y 5 Example with G = Z / 5 Z , A = { 1 , 3 }
Historical Background Rainbow Tur´ an Problem Our Results B ∗ k -sets: Connection to Rainbow Cycles Bipartite Cayley graph construction: B ∗ k -sets → rainbow- C 2 k -free bipartite graphs Construct bipartite graph on X ∪ Y , where X , Y = G Edge ( x , y ) iff y − x = a ∈ A ⊂ G , colour a x 1 y 1 x 2 y 2 y 3 x 3 x 4 y 4 x 5 y 5 Example with G = Z / 5 Z , A = { 1 , 3 }
Recommend
More recommend
