Ramsey properties of random graphs Rajko Nenadov Monash University - PowerPoint PPT Presentation

Ramsey properties of random graphs Rajko Nenadov Monash University 29th August 2016

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H K 6 → K 3

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H K 6 → K 3

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H K 6 → K 3

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H K 6 → K 3

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H K 6 → K 3

Introduction A graph G is Ramsey for H G → H if every red/blue colouring of the edges of G contains a monochromatic copy of H Ramsey (1930) For every graph H there exists (su ffi ciently large) n ∈ N such that K n → H

Introduction A graph G is Ramsey for H Binomial random graph G ( n, p ) • n vertices G → H • each edge present with probability p if every red/blue colouring of the edges of G contains a monochromatic copy of H

Introduction A graph G is Ramsey for H Binomial random graph G ( n, p ) • n vertices G → H • each edge present with probability p if every red/blue colouring of the edges of G contains a monochromatic copy of H Given a graph H and p = p ( n ) ∈ [ 0 , 1 ] , determine Pr[ G ( n, p ) → H ]

Behaviour of Pr[ G ( n, p ) → H ] Given a graph H and p = p ( n ) ∈ [ 0 , 1 ] , determine Pr[ G ( n, p ) → H ]

Behaviour of Pr[ G ( n, p ) → H ] Given a graph H and p = p ( n ) ∈ [ 0 , 1 ] , determine Pr[ G ( n, p ) → H ] • “Being Ramsey for H ” is a monotone property (preserved under edge addition) • Bollob´ as-Thomason (’87): every non-trivial monotone property P has a threshold function p ∗ ( P ) ( 0 , if p/p ∗ ( P ) → 0 n →∞ Pr[ G ( n, p ) ∈ P ] = lim 1 , if p/p ∗ ( P ) → ∞

Behaviour of Pr[ G ( n, p ) → H ] Given a graph H and p = p ( n ) ∈ [ 0 , 1 ] , determine Pr[ G ( n, p ) → H ] • “Being Ramsey for H ” is a monotone property (preserved under edge addition) • Bollob´ as-Thomason (’87): every non-trivial monotone property P has a threshold function p ∗ ( P ) ( 0 , if p/p ∗ ( P ) → 0 n →∞ Pr[ G ( n, p ) ∈ P ] = lim 1 , if p/p ∗ ( P ) → ∞ Goal: find a threshold p H for the property “being Ramsey for H ”

Behaviour of Pr[ G ( n, p ) ! H ] Given a graph H and p = p ( n ) 2 [ 0 , 1 ] , determine Pr[ G ( n, p ) ! H ] • “Being Ramsey for H ” is a monotone property (preserved under edge addition) • Bollob´ as-Thomason (’87): every non-trivial monotone property P has a threshold function p ∗ ( P ) ( 0 , if p ⌧ p ∗ n →∞ Pr[ G ( n, p ) 2 P ] = lim 1 , if p � p ∗ Goal: find a threshold p H for the property “being Ramsey for H ”

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = lim

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 1 < n − 4 / 5 ) • p = n − 1 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = lim

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 1 < n − 4 / 5 ) • p = n − 1 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 0 lim Explanation: G ( n, p ) does not contain K 3

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 1 < n − 4 / 5 ) • p = n − 4 / 5 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = lim

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 1 < n − 4 / 5 ) • p = n − 4 / 5 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 0 lim Explanation: G ( n, p ) does not contain

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 4 / 5 < n − 5 / 7 ) • p = n − 5 / 7 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = lim

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 4 / 5 < n − 5 / 7 ) • p = n − 5 / 7 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 0 lim Explanation: G ( n, p ) does not contain

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 5 / 7 < n − 2 / 3 ) • p = n − 2 / 3 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = lim

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 5 / 7 < n − 2 / 3 ) • p = n − 2 / 3 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 0 lim Explanation: G ( n, p ) does not contain

Warm-up Given a graph H , determine p H ( n ) such that ( 0 , if p ⌧ p H n →∞ Pr[ G ( n, p ) ! H ] = lim 1 , if p � p H Let H = K 3 • p = n − 6 / ( 6 2 ) + ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 1 lim Explanation: G ( n, p ) contains K 6 with high probability ( n − 2 / 3 < n − 1 / 2 ) • p = n − 1 / 2 − ε n →∞ Pr[ G ( n, p ) ! K 3 ] = 0 lim Explanation: whiteboard

Threshold for G ( n, p ) → K 3 Frankl-R¨ odl (’86), Luczak-Ruci´ nski-Voigt (’92’) There exist constants c, C > 0 such that ( if p ≤ cn − 1 / 2 0 , n →∞ Pr[ G ( n, p ) → K 3 ] = lim if p ≥ Cn − 1 / 2 1 ,

Threshold for G ( n, p ) → H R¨ odl-Ruci´ nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that ( if p ≤ cn − β ( H ) 0 , n →∞ Pr[ G ( n, p ) → H ] = lim if p ≥ Cn − β ( H ) 1 ,

Threshold for G ( n, p ) → H R¨ odl-Ruci´ nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that ( if p ≤ cn − β ( H ) 0 , n →∞ Pr[ G ( n, p ) → H ] = lim if p ≥ Cn − β ( H ) 1 , Intuition: β ( H ) is chosen such that • p ≤ cn − β ( H ) → most of the edges do not belong to a copy of H • p ≥ Cn − β ( H ) → each edge belongs to many copies of H

Threshold for G ( n, p ) → H R¨ odl-Ruci´ nski (’93–’95) For every graph H (which contains a cycle) there exist constants c, C > 0 such that ( if p ≤ cn − β ( H ) 0 , n →∞ Pr[ G ( n, p ) → H ] = lim if p ≥ Cn − β ( H ) 1 , Intuition: β ( H ) is chosen such that • p ≤ cn − β ( H ) → most of the edges do not belong to a copy of H • p ≥ Cn − β ( H ) → each edge belongs to many copies of H N.-Steger (2015) – a ‘short’ proof

n →∞ Pr[ G ( n, p ) → K 3 ] = 1 for p ≥ Cn − 1 / 2 lim

n →∞ Pr[ G ( n, p ) → K 3 ] = 1 for p ≥ Cn − 1 / 2 lim or ‘A short introduction to hypergraph containers’