Erd˝ os–Rothschild for Intersecting Families Dennis Clemens 1 Shagnik Das 2 Tuan Tran 2 1 Technische Universit¨ at Hamburg–Harburg 2 Freie Universit¨ at Berlin British Mathematical Colloquium, Bristol 23rd March 2016
Erd˝ os–Rothschild for Intersecting Families Dennis Clemens 1 Shagnik Das 2 , 3 Tuan Tran 2 1 Technische Universit¨ at Hamburg–Harburg 2 Freie Universit¨ at Berlin 3 (I’m this guy) British Mathematical Colloquium, Bristol 23rd March 2016
Introduction A little mathematics Conclusion {} : Outline {} Introductory waffling { ER } The Erd˝ os–Rothschild problem { IF } Intersecting families { ER , IF } Erd˝ os–Rothschild for intersecting families
Introduction A little mathematics Conclusion {} : Outline {} Introductory waffling { ER } The Erd˝ os–Rothschild problem { IF } Intersecting families { ER , IF } Erd˝ os–Rothschild for intersecting families
Introduction A little mathematics Conclusion {} : Outline {} Introductory waffling { ER } The Erd˝ os–Rothschild problem { IF } Intersecting families { ER , IF } Erd˝ os–Rothschild for intersecting families
Introduction A little mathematics Conclusion {} : Outline {} Introductory waffling { ER } The Erd˝ os–Rothschild problem { IF } Intersecting families { ER , IF } Erd˝ os–Rothschild for intersecting families
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � The largest n-vertex triangle-free graph has edges. Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � ex ( n , K 3 ) = . Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � ex ( n , K 3 ) = . Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � ex ( n , K 3 ) = . Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � ex ( n , K 3 ) = . Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion {} : Mantel’s Theorem Theorem (Mantel, 1907) n 2 / 4 � � ex ( n , K 3 ) = . Extensions Tur´ an bounds edges for graphs without larger cliques Stability large triangle-free graphs are close to bipartite Supersaturation number of triangles in larger graphs Mantel: tight
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have?
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Bipartite graph: any two-colouring works
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Bipartite graph: any two-colouring works
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Bipartite graph: any two-colouring works
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Extra edge: causes problems
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Extra edge: causes problems
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Extra edge: causes problems
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Extra edge: causes problems
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Extra edge: causes problems
Introduction A little mathematics Conclusion { ER } : a new extension Question How many two-colourings of its edges without monochromatic triangles can an n-vertex graph have? Conjecture (Erd˝ os–Rothschild, 1974) At most 2 ex ( n , K 3 ) .
Introduction A little mathematics Conclusion { ER } : the known results Theorem (Yuster, 1996) If n is large enough, an n-vertex graph can have at most 2 ex ( n , K 3 ) two-colourings without a monochromatic triangle. Theorem (Alon–Balogh–Keevash–Sudakov, 2004) Given r ∈ { 2 , 3 } , k ≥ 2 and n large enough, an n-vertex graph can have at most r ex ( n , K k ) r-colourings without a monochromatic K k . Further results ◮ When r ≥ 4, maximum is greater than r ex ( n , K k ) ◮ Pikhurko–Yilma (2012): precise answer for r = 4, k ∈ { 3 , 4 } and n large
Introduction A little mathematics Conclusion { IF } : the Erd˝ os–Ko–Rado Theorem Definition � [ n ] � A k -uniform set family F ⊆ is said to be intersecting k if F 1 ∩ F 2 � = ∅ for all F 1 , F 2 ∈ F . Theorem (Erd˝ os–Ko–Rado, 1961) � [ n ] � If n ≥ 2 k and F ⊆ is intersecting, k � n − 1 � 1 |F| ≤ . k − 1 Star with centre 1
Introduction A little mathematics Conclusion { IF } : the Erd˝ os–Ko–Rado Theorem Definition � [ n ] � A k -uniform set family F ⊆ is said to be intersecting k if F 1 ∩ F 2 � = ∅ for all F 1 , F 2 ∈ F . Theorem (Erd˝ os–Ko–Rado, 1961) � [ n ] � If n ≥ 2 k and F ⊆ is intersecting, k � n − 1 � 1 |F| ≤ . k − 1 Star with centre 1
Introduction A little mathematics Conclusion { IF } : the Erd˝ os–Ko–Rado Theorem Definition � [ n ] � A k -uniform set family F ⊆ is said to be t -intersecting k if | F 1 ∩ F 2 | ≥ t for all F 1 , F 2 ∈ F . Theorem (Erd˝ os–Ko–Rado, 1961; Frankl, 1978; Wilson, 1984) � [ n ] � If n ≥ ( t + 1)( k − t + 1) and F ⊆ k is t-intersecting, T � n − t � |F| ≤ . t -star with centre T k − t
Introduction A little mathematics Conclusion { ER , IF } : the Erd˝ os–Rothschild extension Definition An ( r , t )-colouring of a family F is an r -colouring of its members such that each colour class is t -intersecting. Question What is the maximum number of ( r , t ) -colourings a family can have? t-star: r ( n − t k − t )
Introduction A little mathematics Conclusion { ER , IF } : the Erd˝ os–Rothschild extension Definition An ( r , t )-colouring of a family F is an r -colouring of its members such that each colour class is t -intersecting. Question What is the maximum number of ( r , t ) -colourings a family can have? t-star: r ( n − t k − t )
Introduction A little mathematics Conclusion { ER , IF } : the Erd˝ os–Rothschild extension Definition An ( r , t )-colouring of a family F is an r -colouring of its members such that each colour class is t -intersecting. Question What is the maximum number of ( r , t ) -colourings a family can have? t-star: r ( n − t k − t )
Introduction A little mathematics Conclusion { ER , IF } : the Erd˝ os–Rothschild extension Definition An ( r , t )-colouring of a family F is an r -colouring of its members such that each colour class is t -intersecting. Question What is the maximum number of ( r , t ) -colourings a family can have? t-star: r ( n − t k − t )
Introduction A little mathematics Conclusion { ER , IF } : the Erd˝ os–Rothschild extension Definition An ( r , t )-colouring of a family F is an r -colouring of its members such that each colour class is t -intersecting. Question What is the maximum number of ( r , t ) -colourings a family can have? t-star: r ( n − t k − t )
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