Jumps and Non-jumps in q -Multigraphs Steve La Fleur (Joint work - PowerPoint PPT Presentation

Preliminary Known Results Jumps and Non-jumps in q -Multigraphs Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) May 14, 2011 Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Density Definition Given a graph G = ( V , E ) the density of G is defined as d ( G ) = | E | � | V | � 2 Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Erd¨ os-Stone Theorem Theorem Suppose that ε > 0 and ℓ, m are fixed positive integers. Let G be a 1 graph on n vertices with d ( G ) ≥ 1 − ℓ − 1 + ε . If n > n 0 ( ℓ, m , ε ) then G contains a subgraph isomorphic the complete ℓ partite graph with parts of size m, T ℓ, m ℓ . Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Erd¨ os-Stone Theorem Theorem Suppose that ε > 0 and ℓ, m are fixed positive integers. Let G be a 1 graph on n vertices with d ( G ) ≥ 1 − ℓ − 1 + ε . If n > n 0 ( ℓ, m , ε ) then G contains a subgraph isomorphic the complete ℓ partite graph with parts of size m, T ℓ, m ℓ . 1 d ( G ) > 1 − ℓ − 1 and the order of G “large enough” ⇓ G contains a “still large” subgraph with density at least 1 − 1 ℓ Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Let { G n } ∞ n =1 be a sequence of graphs such that | V n | → ∞ as n → ∞ . Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Let { G n } ∞ n =1 be a sequence of graphs such that | V n | → ∞ as n → ∞ . The maximum density of a subgraph on k vertices is given by σ k ( { G n } ) = max max d ( G n [ V ]) n V ∈ ( Vn k ) Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Upper Density Definition The upper density of a sequence of graphs { G n } , denoted as d ( { G n } ) is given by d ( { G n } ) = lim k →∞ σ k Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Examples Consider the sequence of complete, bipartite graphs G n = K n , n Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Examples Consider the sequence of complete, bipartite graphs G n = K n , n ⌊ k 2 ⌋⌈ k 2 ⌉ = 1 σ k ( { K n , n } ) = 2 + o (1) ( k 2 ) Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Examples Consider the sequence of complete, bipartite graphs G n = K n , n ⌊ k 2 ⌋⌈ k 2 ⌉ = 1 σ k ( { K n , n } ) = 2 + o (1) ( k 2 ) d { K n , n } = 1 / 2 Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Examples Consider the sequence of complete, bipartite graphs G n = K n , n ⌊ k 2 ⌋⌈ k 2 ⌉ = 1 σ k ( { K n , n } ) = 2 + o (1) ( k 2 ) d { K n , n } = 1 / 2 More generally, for a fixed integer ℓ consider the sequence of complete, balanced, ℓ -partite graphs G n = T ℓ, n d ( { G n } ) = 1 − 1 ℓ Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Examples Consider the sequence of complete, bipartite graphs G n = K n , n ⌊ k 2 ⌋⌈ k 2 ⌉ = 1 σ k ( { K n , n } ) = 2 + o (1) ( k 2 ) d { K n , n } = 1 / 2 More generally, for a fixed integer ℓ consider the sequence of complete, balanced, ℓ -partite graphs G n = T ℓ, n d ( { G n } ) = 1 − 1 ℓ 1 If { G n } is a sequence of graphs with d ( { G n } ) > 1 − ℓ − 1 , E-S ⇒ d ( { G n } ) ≥ d ( { T ℓ, n } ) = 1 − 1 ℓ Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs What is a jump? Definition A number α is a jump if there exists a constant c = c ( α ) such that, given any sequence of graphs { G n } ∞ n =1 if d ( { G n } ) > α then d ( { G n } ) ≥ α + c . Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs What is a jump? Definition A number α is a jump if there exists a constant c = c ( α ) such that, given any sequence of graphs { G n } ∞ n =1 if d ( { G n } ) > α then d ( { G n } ) ≥ α + c . Question Is every α ∈ [0 , 1) a jump for simple graphs? Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs What is a jump? Definition A number α is a jump if there exists a constant c = c ( α ) such that, given any sequence of graphs { G n } ∞ n =1 if d ( { G n } ) > α then d ( { G n } ) ≥ α + c . Question Is every α ∈ [0 , 1) a jump for simple graphs? Answer: Yes! ℓ − 1 ⇒ d ( { G n } ) ≥ 1 − 1 1 E-S: d ( { G n } ) > 1 − ℓ Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Harder Questions Question Is every number in [0 , 1) a jump for r-uniform hypergraphs? Erd˝ os conjectured that the answer was yes and offered $1000 for a solution. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Harder Questions Question Is every number in [0 , 1) a jump for r-uniform hypergraphs? Erd˝ os conjectured that the answer was yes and offered $1000 for a solution. Frankl and R¨ odl showed that the conjecture was false. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Harder Questions Question Is every number in [0 , 1) a jump for r-uniform hypergraphs? Erd˝ os conjectured that the answer was yes and offered $1000 for a solution. Frankl and R¨ odl showed that the conjecture was false. Question Is every number a jump for multigraphs of bounded multiplicity? Again Erd˝ os conjectured that the answer was yes. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

Preliminary Simple Graphs Known Results q -multigraphs Definitions Definition A q -multigraph G , is a multigraph with edge multiplicity bounded by q . | E | Density = d ( G ) = 2 ) ∈ [0 , q ]. ( | V | σ k = max max d ( G [ V n ]) n V ∈ ( Vn k ) Upper density = d ( { G n } ) = lim k →∞ σ k Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

On [0 , 1) Preliminary On [1 , 2) Known Results For [q-1,q) with q ≥ 4 [0,1): Everything is a jump! Proposition Every number in [0 , 1) is a jump for q-multigraphs for all values of q. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

On [0 , 1) Preliminary On [1 , 2) Known Results For [q-1,q) with q ≥ 4 [0,1): Everything is a jump! Proposition Every number in [0 , 1) is a jump for q-multigraphs for all values of q. For q = 1 this result follows from the Erd¨ os-Stone theorem. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

On [0 , 1) Preliminary On [1 , 2) Known Results For [q-1,q) with q ≥ 4 For q ≥ 2 If G n contains ε n 2 edges of multiplicity q ≥ 2 E-S implies G n contains a large complete bipartite graph with edge multiplicity q . Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

On [0 , 1) Preliminary On [1 , 2) Known Results For [q-1,q) with q ≥ 4 For q ≥ 2 If G n contains ε n 2 edges of multiplicity q ≥ 2 E-S implies G n contains a large complete bipartite graph with edge multiplicity q . If G n contains o ( n 2 ) edges of multiplicity q , then we can remove them without affecting the upper density. Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs

On [0 , 1) Preliminary On [1 , 2) Known Results For [q-1,q) with q ≥ 4 For q ≥ 2 If G n contains ε n 2 edges of multiplicity q ≥ 2 E-S implies G n contains a large complete bipartite graph with edge multiplicity q . If G n contains o ( n 2 ) edges of multiplicity q , then we can remove them without affecting the upper density. Conclusion: Edges of higher multiplicity don’t affect the upper density in any non-trivial way on the interval [0 , 1). Steve La Fleur (Joint work with Paul Horn and Vojtˇ ech R¨ odl) Jumps and Non-jumps in q -Multigraphs