Chapter 2: Method of Alterations The Probabilistic Method Summer 2020 Freie Universität Berlin
Chapter Overview • Introduce the method of alterations
§1 Ramsey Revisited Chapter 2: Method of Alterations The Probabilistic Method
Asymmetric Ramsey Bounds Definition 1.5.4 (Asymmetric Ramsey numbers) Given ℓ, 𝑙 ∈ ℕ , 𝑆(ℓ, 𝑙) is the minimum 𝑜 for which any 𝑜 -vertex graph contains either a clique on ℓ vertices or an independent set on 𝑙 vertices. Obtained lower bounds by considering the random graph 𝐻 𝑜, 𝑞 Corollary 1.5.8 For fixed ℓ ∈ ℕ and 𝑙 → ∞ , we have ℓ−1 𝑙 2 = 𝑆 ℓ, 𝑙 = 𝑃 𝑙 ℓ−1 . Ω 2 ln 𝑙
Triangle-free Graphs Corollary 1.5.8 For fixed ℓ ∈ ℕ and 𝑙 → ∞ , we have ℓ−1 𝑙 2 = 𝑆 ℓ, 𝑙 = 𝑃 𝑙 ℓ−1 . Ω 2 ln 𝑙 The case ℓ = 3 𝑙 • General lower bound gives Ω ln 𝑙 • 𝑆 3, 𝑙 ≥ 𝑙 is utterly trivial • Complete bipartite graph gives 𝑆 3, 𝑙 ≥ 2𝑙 − 1 • Can improve lower bound by more careful computation
Sharper Analysis Theorem 1.5.7 ( ℓ = 3 ) Given 𝑙, 𝑜 ∈ ℕ and 𝑞 ∈ 0,1 , if 𝑜 3 𝑞 3 + 𝑜 𝑙 2 < 1, 1 − 𝑞 𝑙 then 𝑆 3, 𝑙 > 𝑜 . Better estimates 𝑜𝑞 3 3 𝑞 3 ≈ 𝑜 • 6 𝑙 ≈ 2 𝐼 𝑙 𝑜 𝑜 𝑜 • −𝑞𝑙2 𝑙 2 ≈ 𝑓 • 1 − 𝑞 2 ⇒ 𝑆 3, 𝑙 > 𝑑𝑙 for 𝑑 ≈ 1.298
Can We Go Further? What happens for larger 𝑜 ? 𝑙 = 𝑓 𝑙 ln 𝑜 𝑜 𝑜 • 𝑙 ≥ 𝑙 𝑙 2 > 𝑓 −2𝑞 𝑙 𝑙 2 > 𝑓 −𝑞𝑙 2 • 1 − 𝑞 𝑙 • ⇒ need 𝑞 = Ω 𝑙 −1 ln 𝑜 𝑙 , otherwise 𝑜 2 grows exponentially 1 − 𝑞 𝑙 3 3 𝑞 3 = Θ 𝑜𝑞 3 = Θ 𝑜 𝑜 • But then 𝑜 𝑙 ln 𝑙 • Bigger than 1 if 𝑜 > 𝐷𝑙 for some constant 𝐷 Lemma 2.1.1 There is some constant 𝐷 such that, if 𝑜 > 𝐷𝑙 , then 𝑜 3 𝑞 3 + 𝑜 𝑙 2 > 1. 1 − 𝑞 𝑙
Reinterpreting the Proof Proof we saw 𝑙 3 𝑞 3 + 𝑜 𝑜 • ℙ 𝐻 𝑜, 𝑞 not Ramsey ≤ 1 − 𝑞 2 𝑙 • If this is less than 1 , we get a Ramsey graph with positive probability • If this is more than 1 , we get no useful information Linearity of expectation 𝑜 • 3 𝑞 3 : expected number of triangles in 𝐻 𝑜, 𝑞 𝑙 𝑜 2 : expected number of independent sets of size 𝑙 in 𝐻 𝑜, 𝑞 • 1 − 𝑞 𝑙 𝑙 3 𝑞 3 + 𝑜 𝑜 2 is the expected number of bad subgraphs • ⇒ 1 − 𝑞 𝑙 • If this is less than 1 , then with positive probability we have no bad subgraphs • ⇒ we get a Ramsey graph
Shades of Grey Great expectations • What does 𝔽 # bad subgraphs ≥ 1 mean? • Do we have to have bad subgraphs? • Not necessarily; see Chapter 3 for details • Gives some guarantee of goodness 𝑙 3 𝑞 3 + • There is a graph with at most 𝑜 𝑜 2 bad subgraphs 1 − 𝑞 𝑙 • If this is small, perhaps we can fix it Method of Alterations Goal: existence of an object with property 𝒬 1. Show random object is with positive probability close to having 𝒬 2. Make deterministic changes to the random object to achieve 𝒬
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets
Graph Surgery Given • Graph with few triangles/large independent sets Goal • Edit graph to obtain a Ramsey graph Idea: remove an edge from each triangle
Graph Surgery Given • Graph with few triangles/large independent sets Goal • Edit graph to obtain a Ramsey graph Idea: remove an edge from each triangle
Graph Surgery Given • Graph with few triangles/large independent sets Goal • Edit graph to obtain a Ramsey graph Idea: remove an edge from each triangle Problem: creates new independent sets
Graph Surgery Given • Graph with few triangles/large independent sets Goal • Edit graph to obtain a Ramsey graph Solution: remove a vertex from each triangle/independent set
Graph Surgery Given • Graph with few triangles/large independent sets Goal • Edit graph to obtain a Ramsey graph Solution: remove a vertex from each triangle/independent set Result: a Ramsey graph, albeit on fewer vertices
An Altered Theorem Theorem 2.1.2 For every 𝑜, ℓ, 𝑙 ∈ ℕ and 𝑞 ∈ [0,1] , we have 𝑆 ℓ, 𝑙 > 𝑜 − 𝑜 2 − 𝑜 ℓ 𝑙 2 . ℓ 𝑞 1 − 𝑞 𝑙 Proof • Let 𝐻 ∼ 𝐻 𝑜, 𝑞 ℓ 𝑙 𝑜 2 + 𝑜 2 is the expected number of 𝐿 ℓ and 𝐿 𝑙 • 𝜈 ≔ ℓ 𝑞 1 − 𝑞 𝑙 • ⇒ there is an 𝑜 -vertex graph with at most 𝜈 bad subgraphs • Delete one vertex from each bad subgraph • Obtain a Ramsey subgraph on at least 𝑜 − 𝜈 vertices ∎
𝑆(3, 𝑙) : A New Bound Theorem 2.1.2 ( ℓ = 3 ) For every 𝑜, 𝑙 ∈ ℕ and 𝑞 ∈ [0,1] , we have 𝑆 3, 𝑙 > 𝑜 − 𝑜 3 𝑞 3 − 𝑜 𝑙 2 . 1 − 𝑞 𝑙 Goal 𝑙 3 𝑞 3 − 𝑜 𝑜 • Choose 𝑜, 𝑞 to maximise 𝑜 − 1 − 𝑞 2 𝑙 Choosing 𝑞 • Small 𝑞 makes the second term small 𝑜 • Recall: need 𝑞 = Ω 𝑙 −1 ln 𝑙 , otherwise third term exponentially large • When 𝑞 is this large, third term exponentially small – insignificant
𝑆(3, 𝑙) : A New Bound Recall 𝑙 3 𝑞 3 − 𝑜 𝑜 • Maximising 𝑜 − 1 − 𝑞 2 𝑙 • Take 𝑞 = Θ 𝑙 −1 ln 𝑜 𝑙 Choosing 𝑜 3 𝑜 𝑜 • Want to maximise 𝑜 − Θ 𝑙 ln 𝑙 3 𝑜 𝑜 • At maximum: 𝑙 ln = Θ 𝑜 𝑙 3 3 2 𝑙 𝑙 2 • ⇒ 𝑜 = Θ = Θ ln 𝑜 ln 𝑙 𝑙
Where We Stand Corollary 2.1.3 As 𝑙 → ∞ , we have 3 2 𝑙 R 3, k = Ω . ln 𝑙 Lower bound • Superlinear lower bound • Beats Turán Upper bound • Erd ő s-Szekeres: 𝑆 3, 𝑙 = 𝑃(𝑙 2 ) • Can we narrow the gap? Stay tuned!
Any questions?
§2 Dominating Sets Chapter 2: Method of Alterations The Probabilistic Method
BER: A Modern Tragicomedy Sep 2006 Berlin-Brandenburg Airport to open Oct 2011 Jun 2010 Opening postponed to Jun 2012 May 2012 Fire detection systems do not work! Solution • Hire people to stand around the airport looking for signs of fire Problem • Already overbudget • ⇒ want to hire as few people as possible
Combinatorics to the Rescue The airport is a graph • Vertices: areas where fire could break out • Edges: lines of sight between areas Objective • Find a set of vertices that “see” all other vertices
Combinatorics to the Rescue The airport is a graph • Vertices: areas where fire could break out • Edges: lines of sight between areas Objective • Find a set of vertices that “see” all other vertices
Combinatorics to the Rescue The airport is a graph • Vertices: areas where fire could break out • Edges: lines of sight between areas Objective • Find a set of vertices that “see” all other vertices
Combinatorics to the Rescue The airport is a graph • Vertices: areas where fire could break out • Edges: lines of sight between areas Objective • Find a set of vertices that “see” all other vertices
Small Dominating Sets Definition 2.2.1 Given a graph 𝐻 = 𝑊, 𝐹 , a set 𝑇 ⊆ 𝑊 of vertices is a dominating set if, for every 𝑤 ∈ 𝑊 ∖ 𝑇 , there is some 𝑡 ∈ 𝑇 with 𝑡, 𝑤 ∈ 𝐹 . Extremal problem • How large can the smallest dominating set of an 𝑜 -vertex graph 𝐻 be? Answer • 𝑜 (!) • Isolated vertices must be in any dominating set Avoiding trivialities • What if we require 𝐻 to have minimum degree 𝜀 ?
Do Random Sets Dominate? Problem Given 𝐻 on 𝑜 vertices with 𝜀 𝐻 ≥ 𝜀 , how large can its smallest dominating set be? Random set • Let 𝑇 ⊆ 𝑊 be a random set • 𝑤 ∈ 𝑇 with probability 𝑞 , independently Undominated vertices • For 𝑣 ∈ 𝑊 , define the event 𝐹 𝑣 = 𝑣 not dominated by 𝑇 • For 𝐹 𝑣 to hold, need: • 𝑣 ∉ 𝑇 • 𝑤 ∉ 𝑇 for all neighbours 𝑤 of 𝑣 • ⇒ ℙ 𝐹 𝑣 = 1 − 𝑞 𝑒 𝑣 +1
Calculations Continued Failure probability • 𝑇 not dominating = ∪ 𝑣∈𝑊 𝐹 𝑣 • ℙ ∪ 𝑣∈𝑊 𝐹 𝑣 < σ 𝑣∈𝑊 ℙ 𝐹 𝑣 = σ 𝑣∈𝑊 1 − 𝑞 𝑒 𝑣 +1 • σ 𝑣∈𝑊 1 − 𝑞 𝑒 𝑣 +1 ≤ 𝑜 1 − 𝑞 𝜀+1 ≤ 𝑜𝑓 −𝑞 𝜀+1 ln 𝑜 • ⇒ if 𝑞 = 𝜀+1 , then ℙ 𝑇 not dominating < 1 • ⇒ 𝑇 is dominating with positive probability Size of the dominating set • 𝑇 ∼ Bin 𝑜, 𝑞 𝑜 ln 𝑜 • ⇒ 𝔽 𝑇 = 𝑜𝑞 = 𝜀+1 𝑜 ln 𝑜 • ⇒ with positive probability, 𝑇 ≤ 𝜀+1
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