Chapter 6: The Rdl Nibble The Probabilistic Method Summer 2020 - PowerPoint PPT Presentation

Chapter 6: The Rödl Nibble The Probabilistic Method Summer 2020 Freie Universität Berlin

Chapter Overview • Introduce the Erd ős -Hanani Conjecture • Prove it with the Rödl Nibble

§1 The Erd ő s-Hanani Conjecture Chapter 6: The Rödl Nibble The Probabilistic Method

Edge-disjoint Triangles Recall • Bounding the probability of 𝐻 𝑜, 𝑞 being 𝐿 3 -free • Restricted our attention to mutually independent events • ⇔ edge-disjoint triangles Lemma 5.4.1 1 𝑜−1 There exists a family of pairwise edge-disjoint triangles in 𝐿 𝑜 . 2 3 Larger cliques • Can run the same argument for the probability of being 𝐿 𝑙 -free • Want to find a large collection of edge-disjoint 𝑙 -cliques

Hypergraphs and Packings “Graphs are for babies” - Tom Trotter, 2017 Random 𝑢 -uniform hypergraph 𝐼 𝑢 𝑜, 𝑞 • Vertex set 𝑊 = 𝑜 𝑜 • Edges: each 𝑢 -set in an edge independently with probability 𝑞 𝑢 Clique containment 𝑢 ⊆ 𝐼 𝑢 𝑜, 𝑞 • Can ask for threshold for 𝐿 𝑙 • Upper bound on probability: use edge-disjoint hypercliques Definition 6.1.1 (Packings) 𝑜 A (𝑙, 𝑢) -packing in [𝑜] is a family of 𝑙 -sets ℱ ⊆ such that every 𝑢 - 𝑙 set is contained in at most one member of the family.

An Extremal Problem Maximum packings • For effective bounds, want as large a packing as possible • 𝑛 𝑜, 𝑙, 𝑢 = max ℱ ∶ ℱ is a 𝑙, 𝑢 −packing on 𝑜 Proposition 6.1.2 𝑜 𝑢 For all 𝑜 ≥ 𝑙 ≥ 𝑢 , we have 𝑛 𝑜, 𝑙, 𝑢 ≤ . 𝑙 𝑢 Proof 𝐺 • Given packing ℱ , double-count pairs (𝐺, 𝑈) with 𝐺 ∈ ℱ and 𝑈 ∈ 𝑢 • Each 𝐺 ∈ ℱ has 𝑙 𝑙 𝑢 subsets of size 𝑢 ⇒ ℱ 𝑢 pairs • Each 𝑢 -set covered at most once ⇒ at most 𝑜 𝑢 pairs ∎

The Case of Equality Proposition 6.1.2 𝑜 𝑢 For all 𝑜 ≥ 𝑙 ≥ 𝑢 , we have 𝑛 𝑜, 𝑙, 𝑢 ≤ . 𝑙 𝑢 Remarks 1 1 𝑜−1 𝑜 • With our earlier construction, shows ≤ 𝑛 𝑜, 3,2 ≤ 2 2 3 3 • Can we do better? • Tightness in Proposition 6.1.2: every 𝑢 -set covered exactly once Definition 6.1.3 (Designs) 𝑜 A 𝑢 - 𝑜, 𝑙, 1 design is a family of 𝑙 -sets ℱ ⊆ such that every 𝑢 -set 𝑙 𝑜 𝑈 ∈ is contained in exactly one set 𝐺 ∈ ℱ . 𝑢

The Utility of Designs Definition 6.1.3 (Designs) 𝑜 A 𝑢 - 𝑜, 𝑙, 1 design is a family of 𝑙 -sets ℱ ⊆ such that every 𝑢 -set 𝑙 𝑜 𝑈 ∈ is contained in exactly one set 𝐺 ∈ ℱ . 𝑢 Useful objects • Study originated in field of experiment design Examples ( 𝑙 = 3 , 𝑢 = 2 ) 𝑜 = 7 𝑜 = 9

Divisibility Restrictions Proposition 6.1.4 If a 𝑢 - (𝑜, 𝑙, 1) design exists, then, for every 0 ≤ 𝑗 ≤ 𝑢 − 1 , 𝑜−𝑗 𝑢−𝑗 is divisible by 𝑙−𝑗 𝑢−𝑗 . Proof 𝑜 • Fix a design ℱ ⊆ 𝑙 , and consider 𝑗 ⊆ 𝑜 • There are 𝑜−𝑗 𝑢−𝑗 𝑢 -sets 𝑈 with 𝑗 ⊆ 𝑈 • Each such 𝑈 is contained in exactly one set 𝐺 ∈ ℱ • Each such 𝐺 contains 𝑙−𝑗 𝑢−𝑗 𝑢 -sets 𝑈 with 𝑗 ⊆ 𝑈 𝑙−𝑗 𝑜−𝑗 • ⇒ ℱ = ∎ 𝑢−𝑗 𝑢−𝑗 • e.g.: a 2 - 𝑜, 3,1 design can only exist when 𝑜 ≡ 1,3 mod 6

Asymptotic Designs Difficulties • Probabilistic method is blind to arithmetic conditions • Suggests designs will be hard to construct Approximation • How large a packing can we find? • Can we ensure that almost all 𝑢 -sets are contained in a 𝑙 -set from the family? Conjecture 6.1.5 (Er dő s-Hanani, 1963) For fixed 𝑙 ≥ 𝑢 ≥ 1 , as 𝑜 → ∞ , we have 𝑜 𝑢 𝑛 𝑜, 𝑙, 𝑢 = 1 − 𝑝 1 . 𝑙 𝑢

A Dual Problem Types of set families • 𝑙, 𝑢 -packings: 𝑙 -sets that cover every 𝑢 -set at most once • 𝑢 - (𝑜, 𝑙, 1) designs: 𝑙 -sets that cover every 𝑢 -set exactly once Definition 6.1.6 (Coverings) 𝑜 A 𝑙, 𝑢 -covering of 𝑜 is a family of 𝑙 -sets ℱ ⊆ such that every 𝑢 - 𝑙 𝑜 set 𝑈 ∈ is contained in at least one set 𝐺 ∈ ℱ . The size of the 𝑢 smallest 𝑙, 𝑢 -covering of 𝑜 is denoted by 𝑁 𝑜, 𝑙, 𝑢 . Proposition 6.1.7 𝑜 𝑢 For all 𝑜 ≥ 𝑙 ≥ 𝑢 , we have 𝑁 𝑜, 𝑙, 𝑢 ≥ . 𝑙 𝑢

Asymptotic Packings and Coverings Proposition 6.1.8 For fixed 𝑙 ≥ 𝑢 , we have 𝑙 𝑙 𝑛 𝑜, 𝑙, 𝑢 𝑁 𝑜, 𝑙, 𝑢 𝑢 𝑢 lim = 1 ⇔ lim = 1. 𝑜 𝑜 𝑜→∞ 𝑜→∞ 𝑢 𝑢 Proof ⇒ 𝑜 • Let ℱ be a 𝑙, 𝑢 -packing of size 1 − o 1 𝑢 𝑙 𝑢 𝑙 𝑜 • Then ℱ covers ℱ = 1 − 𝑝 1 𝑢 of the 𝑢 -sets 𝑢 • Form a cover ℱ ′ by adding a 𝑙 -set covering each uncovered 𝑢 -set 𝑜 𝑜 • ℱ ′ = 1 − 𝑝 1 𝑜 𝑢 𝑢 + 𝑝 1 = 1 + 𝑝 1 ∎ 𝑙 𝑢 𝑙 𝑢 𝑢

Asymptotic Packings and Coverings Proposition 6.1.8 For fixed 𝑙 ≥ 𝑢 , we have 𝑙 𝑙 𝑛 𝑜, 𝑙, 𝑢 𝑁 𝑜, 𝑙, 𝑢 𝑢 𝑢 lim = 1 ⇔ lim = 1. 𝑜 𝑜 𝑜→∞ 𝑜→∞ 𝑢 𝑢 Proof ⇐ 𝑜 • Let ℱ be a 𝑙, 𝑢 -covering of size 1 + o 1 𝑢 𝑙 𝑢 • For each 𝑢 -set 𝑈 , let 𝑒 𝑈 = 𝐺 ∈ ℱ: 𝑈 ⊆ 𝐺 be its degree in ℱ • Form a 𝑙, 𝑢 -packing ℱ ′ by deleting for each 𝑢 -set 𝑈 any excess covering sets 𝑜 𝑙 𝑜 𝑜 • # deleted sets ≤ σ 𝑈 𝑒 𝑈 − 1 = σ 𝑈 𝑒 𝑈 − = ℱ − = 𝑝 𝑢 𝑢 𝑢 𝑢 𝑜 • ⇒ ℱ ′ = 1 − 𝑝 1 𝑢 ∎ 𝑙 𝑢

The Random Hypergraph • Does 𝐼 𝑙 (𝑜, 𝑞) form a good cover? Covering sets 𝑜 is contained in 𝑜−𝑢 • A fixed 𝑢 -set 𝑈 ∈ 𝑙−𝑢 sets of size 𝑙 𝑢 𝑜−𝑢 • ⇒ ℙ 𝑈 uncovered by 𝐼 𝑙 𝑜, 𝑞 𝑜−𝑢 𝑙−𝑢 ≥ exp −2𝑞 = 1 − 𝑞 𝑙−𝑢 𝑜 𝑢 exp −2𝑞 𝑜−𝑢 • ⇒ 𝔽 # uncovered 𝑢−sets ≥ 𝑙−𝑢 log 𝑜 • ⇒ to cover all 𝑢 -sets, need 𝑞 = Ω 𝑢 𝑜−𝑢 𝑙−𝑢 Size of cover • 𝐼 𝑙 𝑜, 𝑞 ~ Bin 𝑜 𝑙 , 𝑞 𝑙 log 𝑜 𝑜 𝑜 𝑢 log 𝑜 • ⇒ with high probability, size of cover = Ω 𝑢 𝑢 = Ω 𝑜−𝑢 𝑙 𝑙−𝑢 𝑢

Summary So Far Corollary 6.1.9 𝑜 𝑜 ≤ 𝑁 𝑜, 𝑙, 𝑢 = 𝑃 log 𝑜 𝑢 𝑢 For 𝑙 ≥ 𝑢 , we have . 𝑙 𝑢 𝑙 𝑢 𝑢 Lower bound • Double counting: each 𝑙 -set covers only 𝑙 𝑢 of the 𝑜 𝑢 𝑢 -sets Upper bound • Random hypergraph 𝐼 𝑙 𝑜, 𝑞 of appropriate density Conjecture 6.1.5’ ( Erd ő s-Hanani, 1963) 𝑜 𝑢 For fixed 𝑙 ≥ 𝑢 , as 𝑜 → ∞ , we have 𝑁 𝑜, 𝑙, 𝑢 = 1 + 𝑝 1 . 𝑙 𝑢

Any questions?

§2 The Nibble Chapter 6: The Rödl Nibble The Probabilistic Method

Rödl to the Rescue Conjecture 6.1.5’ ( Erd ő s-Hanani, 1963) 𝑜 𝑢 For fixed 𝑙 ≥ 𝑢 , as 𝑜 → ∞ , we have 𝑁 𝑜, 𝑙, 𝑢 = 1 + 𝑝 1 . 𝑙 𝑢 Theorem 6.2.1 (Rödl, 1985) The Erd ő s-Hanani Conjecture is true. Generalisation • Rö dl’s objective was to prove the Er dő s-Hanani Conjecture • His method, the Rödl Nibble, applies in more general settings • We shall see a generalisation due to Pippinger (1989)

Hypergraph Covers Definition 6.2.2 (Cover) Let 𝐼 = 𝑊, 𝐹 be an 𝑠 -uniform 𝑜 -vertex hypergraph without isolated vertices. A cover of 𝐼 is a collection of edges ℱ ⊆ 𝐹(𝐼) that covers all the vertices; that is, ∪ 𝑓∈ℱ 𝑓 = 𝑊 𝐼 . Remarks • A cover of 𝐼 is an 𝑜, 𝑠, 1 -covering, whose sets are edges of 𝐼 𝑜 • Each cover must contain at least 𝑠 edges 𝑠 • Trivial to find covers of this size when 𝐼 = 𝐿 𝑜 • Take a maximum matching • If needed, add one edge with remaining vertices • Can we guarantee small covers in sparser hypergraphs?

Pippinger’s Theorem Theorem 6.2.3 (Pippinger, 1989) For every 𝑠 ≥ 2 and large enough 𝐸 ∈ ℕ , any 𝑠 -uniform 𝑜 -vertex hypergraph 𝐼 without isolated vertices that satisfies the following conditions: 1. Almost all vertices have degree approximately 𝐸, 2. All vertices have degree 𝑃 𝐸 , 3. Every pair of vertices have 𝑝 𝐸 common edges, 𝑜 has a cover of size 1 + 𝑝 1 𝑠 .

A Non-example • Bounded degrees and co-degrees are necessary Construction • Consider a star – all edges containing some fixed vertex 𝑤 0 • Almost all vertices have degree 𝑜−2 𝑠−2 𝑜−1 𝑜−2 • But deg 𝑤 0 = 𝑠−1 ≫ 𝑠−2 • Most pairs of vertices have co-degree 𝑜−3 𝑠−3 • However, 𝑤 0 and any other vertex have co-degree 𝑜−2 𝑠−2 Large covers • Each edge covers 𝑠 − 1 vertices from 𝑊 𝐼 ∖ 𝑤 0 𝑜−1 1 𝑜 • ⇒ each cover has size at least 𝑠−1 ≈ 1 + 𝑠−1 𝑠