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 Erdős-Hanani Conjecture
• Prove it with the Rödl Nibble

§1 The Erdős-Hanani Conjecture

Chapter 6: The Rödl Nibble The Probabilistic Method

Lemma 5.4.1 There exists a family of

1 3 𝑜−1 2

pairwise edge-disjoint triangles in 𝐿𝑜.

Edge-disjoint Triangles

Recall

• Bounding the probability of 𝐻 𝑜, 𝑞 being 𝐿3-free
• Restricted our attention to mutually independent events
• ⇔ edge-disjoint triangles

Larger cliques

• Can run the same argument for the probability of being 𝐿𝑙-free
• Want to find a large collection of edge-disjoint 𝑙-cliques

Definition 6.1.1 (Packings) A (𝑙, 𝑢)-packing in [𝑜] is a family of 𝑙-sets ℱ ⊆

𝑜 𝑙

such that every 𝑢- set is contained in at most one member of the family.

Hypergraphs and Packings

“Graphs are for babies” - Tom Trotter, 2017 Random 𝑢-uniform hypergraph 𝐼 𝑢 𝑜, 𝑞

• Vertex set 𝑊 = 𝑜
• Edges: each 𝑢-set in

𝑜 𝑢

an edge independently with probability 𝑞

Clique containment

• Can ask for threshold for 𝐿𝑙

𝑢 ⊆ 𝐼 𝑢 𝑜, 𝑞

• Upper bound on probability: use edge-disjoint hypercliques

Proposition 6.1.2 For all 𝑜 ≥ 𝑙 ≥ 𝑢, we have 𝑛 𝑜, 𝑙, 𝑢 ≤

𝑜 𝑢 𝑙 𝑢

.

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

Proof

• Given packing ℱ, double-count pairs (𝐺, 𝑈) with 𝐺 ∈ ℱ and 𝑈 ∈

𝐺 𝑢

• Each 𝐺 ∈ ℱ has 𝑙

𝑢 subsets of size 𝑢 ⇒ ℱ 𝑙 𝑢 pairs

• Each 𝑢-set covered at most once ⇒ at most 𝑜

𝑢 pairs

∎

Proposition 6.1.2 For all 𝑜 ≥ 𝑙 ≥ 𝑢, we have 𝑛 𝑜, 𝑙, 𝑢 ≤

𝑜 𝑢 𝑙 𝑢

.

The Case of Equality

Remarks

• With our earlier construction, shows

1 3 𝑜−1 2

≤ 𝑛 𝑜, 3,2 ≤

1 3 𝑜 2

• Can we do better?
• Tightness in Proposition 6.1.2: every 𝑢-set covered exactly once

Definition 6.1.3 (Designs) A 𝑢- 𝑜, 𝑙, 1 design is a family of 𝑙-sets ℱ ⊆

𝑜 𝑙

such that every 𝑢-set 𝑈 ∈

𝑜 𝑢

is contained in exactly one set 𝐺 ∈ ℱ.

The Utility of Designs

Useful objects

• Study originated in field of experiment design

Examples (𝑙 = 3, 𝑢 = 2) Definition 6.1.3 (Designs) A 𝑢- 𝑜, 𝑙, 1 design is a family of 𝑙-sets ℱ ⊆

𝑜 𝑙

such that every 𝑢-set 𝑈 ∈

𝑜 𝑢

is contained in exactly one set 𝐺 ∈ ℱ. 𝑜 = 7 𝑜 = 9

Divisibility Restrictions

Proof

• Fix a design ℱ ⊆

𝑜 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

• ⇒ ℱ

𝑙−𝑗 𝑢−𝑗

=

𝑜−𝑗 𝑢−𝑗

∎

• e.g.: a 2- 𝑜, 3,1 design can only exist when 𝑜 ≡ 1,3 mod 6

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 .

Conjecture 6.1.5 (Erdős-Hanani, 1963) For fixed 𝑙 ≥ 𝑢 ≥ 1, as 𝑜 → ∞, we have 𝑛 𝑜, 𝑙, 𝑢 = 1 − 𝑝 1

𝑜 𝑢 𝑙 𝑢

.

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?

Definition 6.1.6 (Coverings) A 𝑙, 𝑢 -covering of 𝑜 is a family of 𝑙-sets ℱ ⊆

𝑜 𝑙

such that every 𝑢- set 𝑈 ∈

𝑜 𝑢

is contained in at least one set 𝐺 ∈ ℱ. The size of the smallest 𝑙, 𝑢 -covering of 𝑜 is denoted by 𝑁 𝑜, 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

Proposition 6.1.7 For all 𝑜 ≥ 𝑙 ≥ 𝑢, we have 𝑁 𝑜, 𝑙, 𝑢 ≥

𝑜 𝑢 𝑙 𝑢

.

Asymptotic Packings and Coverings

Proof ⇒

• Let ℱ be a 𝑙, 𝑢 -packing of size 1 − o 1

𝑜 𝑢 𝑙 𝑢

• Then ℱ covers ℱ

𝑙 𝑢

= 1 − 𝑝 1

𝑜 𝑢 of the 𝑢-sets

• Form a cover ℱ′ by adding a 𝑙-set covering each uncovered 𝑢-set
• ℱ′ = 1 − 𝑝 1

𝑜 𝑢 𝑙 𝑢

+ 𝑝 1

𝑜 𝑢

= 1 + 𝑝 1

𝑜 𝑢 𝑙 𝑢

∎

Proposition 6.1.8 For fixed 𝑙 ≥ 𝑢, we have lim

𝑜→∞

𝑛 𝑜, 𝑙, 𝑢

𝑙 𝑢 𝑜 𝑢

= 1 ⇔ lim

𝑜→∞

𝑁 𝑜, 𝑙, 𝑢

𝑙 𝑢 𝑜 𝑢

= 1.

Asymptotic Packings and Coverings

Proof ⇐

• Let ℱ be a 𝑙, 𝑢 -covering of size 1 + o 1

𝑜 𝑢 𝑙 𝑢

• For each 𝑢-set 𝑈, let 𝑒𝑈 =

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
• # deleted sets ≤ σ𝑈 𝑒𝑈 − 1 = σ𝑈 𝑒𝑈 −

𝑜 𝑢

=

𝑙 𝑢

ℱ −

𝑜 𝑢

= 𝑝

𝑜 𝑢

• ⇒ ℱ′ = 1 − 𝑝 1

𝑜 𝑢 𝑙 𝑢

∎

Proposition 6.1.8 For fixed 𝑙 ≥ 𝑢, we have lim

𝑜→∞

𝑛 𝑜, 𝑙, 𝑢

𝑙 𝑢 𝑜 𝑢

= 1 ⇔ lim

𝑜→∞

𝑁 𝑜, 𝑙, 𝑢

𝑙 𝑢 𝑜 𝑢

= 1.

The Random Hypergraph

• Does 𝐼 𝑙 (𝑜, 𝑞) form a good cover?

Covering sets

• A fixed 𝑢-set 𝑈 ∈

𝑜 𝑢

is contained in 𝑜−𝑢

𝑙−𝑢 sets of size 𝑙

• ⇒ ℙ 𝑈 uncovered by 𝐼 𝑙 𝑜, 𝑞

= 1 − 𝑞

𝑜−𝑢 𝑙−𝑢 ≥ exp −2𝑞

𝑜−𝑢 𝑙−𝑢

• ⇒ 𝔽 # uncovered 𝑢−sets ≥

𝑜 𝑢 exp −2𝑞 𝑜−𝑢 𝑙−𝑢

• ⇒ to cover all 𝑢-sets, need 𝑞 = Ω

log 𝑜

𝑢 𝑜−𝑢 𝑙−𝑢

Size of cover

• 𝐼 𝑙 𝑜, 𝑞 ~ Bin

𝑜 𝑙 , 𝑞

• ⇒ with high probability, size of cover = Ω

𝑜 𝑙 log 𝑜 𝑢 𝑜−𝑢 𝑙−𝑢

= Ω

𝑜 𝑢 log 𝑜 𝑢 𝑙 𝑢

Corollary 6.1.9 For 𝑙 ≥ 𝑢, we have

𝑜 𝑢 𝑙 𝑢

≤ 𝑁 𝑜, 𝑙, 𝑢 = 𝑃 log 𝑜

𝑢

𝑜 𝑢 𝑙 𝑢

.

Summary So Far

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’ (Erdős-Hanani, 1963) For fixed 𝑙 ≥ 𝑢, as 𝑜 → ∞, we have 𝑁 𝑜, 𝑙, 𝑢 = 1 + 𝑝 1

𝑜 𝑢 𝑙 𝑢

.

Any questions?

§2 The Nibble

Chapter 6: The Rödl Nibble The Probabilistic Method

Conjecture 6.1.5’ (Erdős-Hanani, 1963) For fixed 𝑙 ≥ 𝑢, as 𝑜 → ∞, we have 𝑁 𝑜, 𝑙, 𝑢 = 1 + 𝑝 1

𝑜 𝑢 𝑙 𝑢

.

Rödl to the Rescue

Generalisation

• Rödl’s objective was to prove the Erdős-Hanani Conjecture
• His method, the Rödl Nibble, applies in more general settings
• We shall see a generalisation due to Pippinger (1989)

Theorem 6.2.1 (Rödl, 1985) The Erdős-Hanani Conjecture is true.

Hypergraph Covers

Remarks

• A cover of 𝐼 is an 𝑜, 𝑠, 1 -covering, whose sets are edges of 𝐼
• Each cover must contain at least

𝑜 𝑠 edges

• Trivial to find covers of this size when 𝐼 = 𝐿𝑜

𝑠

• Take a maximum matching
• If needed, add one edge with remaining vertices
• Can we guarantee small covers in sparser hypergraphs?

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 .

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,

has a cover of size 1 + 𝑝 1

𝑜 𝑠.

A Non-example

• Bounded degrees and co-degrees are necessary

Construction

• Consider a star – all edges containing some fixed vertex 𝑤0
• Almost all vertices have degree 𝑜−2

𝑠−2

• But deg 𝑤0 =

𝑜−1 𝑠−1 ≫ 𝑜−2 𝑠−2

• Most pairs of vertices have co-degree 𝑜−3

𝑠−3

• However, 𝑤0 and any other vertex have co-degree 𝑜−2

𝑠−2

Large covers

• Each edge covers 𝑠 − 1 vertices from 𝑊 𝐼 ∖ 𝑤0
• ⇒ each cover has size at least

𝑜−1 𝑠−1 ≈ 1 + 1 𝑠−1 𝑜 𝑠

Pippenger’s Precise Theorem

Theorem 6.2.3 (Pippinger, 1989) For every integer 𝑠 ≥ 2 and reals 𝜆 ≥ 1 and 𝛽 > 0, there are 𝛿 = 𝛿 𝑠, 𝜆, 𝛽 > 0 and 𝐸0 = 𝐸0(𝑠, 𝜆, 𝛽) such that for every 𝑜 ≥ 𝐸 ≥ 𝐸0, any 𝑠-uniform 𝑜-vertex hypergraph 𝐼 without isolated vertices that satisfies the following conditions:

• 1. All but at most 𝛿𝑜 vertices have degree 1 ± 𝛿 𝐸,
• 2. All vertices have degree at most 𝜆𝐸,
• 3. Every pair of vertices have co-degree at most 𝛿𝐸,

has a cover of size at most 1 + 𝛽

𝑜 𝑠.

Small Coverings

Proof

• Build an auxiliary 𝑠-graph 𝐼, for 𝑠 ≔

𝑙 𝑢

• 𝑊 𝐼 =

𝑜 𝑢

and 𝐹 𝐼 =

𝐺 𝑢 : 𝐺 ∈ 𝑜 𝑙

• Cover of 𝐼 ↔ (𝑙, 𝑢)-covering of 𝑜
• Hypergraph is 𝐸-regular for 𝐸 ≔

𝑜−𝑢 𝑙−𝑢 ⇒ 𝜆 = 1

• Co-degrees are at most 𝑜− 𝑢+1

𝑙− 𝑢+1

=

𝑙−𝑢 𝑜−𝑢 𝐸 ≤ 𝛿𝐸 when 𝑜 is large

• Satisfy Pippinger’s conditions for any 𝛽
• ⇒ cover (hence covering) of size at most 1 + 𝛽

𝑜 𝑢 𝑙 𝑢

∎

Conjecture 6.1.5’ (Erdős-Hanani, 1963) For fixed 𝑙 ≥ 𝑢, as 𝑜 → ∞, we have 𝑁 𝑜, 𝑙, 𝑢 = 1 + 𝑝 1

𝑜 𝑢 𝑙 𝑢

.

Proving Pippinger

“There is only one way to eat an elephant, a bite at a time.” – Desmond Tutu The failure of randomness

• Cover some vertices several times before covering others
• Fix: prevent the random process from doing so
• Remove covered vertices from consideration

An iterative approach

• Choose a small number of edges at random
• Hope that they are mostly disjoint
• Remove the covered vertices from the hypergraph
• Hope that the remaining edges are still well-distributed
• Repeat until everything is covered

One Step at a Time

Lemma 6.2.4 For every integer 𝑠 ≥ 2 and reals 𝜇 ≥ 1, 𝜁 > 0 and 𝜀′ > 0, there are 𝜀 = 𝜀 𝑠, 𝜇, 𝜁, 𝜀′ and 𝐸0 = 𝐸0 𝑠, 𝜇, 𝜁, 𝜀′ such that, for every 𝑜 ≥ 𝐸 ≥ 𝐸0, every 𝑠-uniform 𝑜-vertex hypergraph 𝐼 = 𝑊, 𝐹 satisfying

• 1. For all vertices 𝑤 ∈ 𝑊 except at most 𝜀𝑜, deg 𝑤 = 1 ± 𝜀 𝐸,
• 2. For all vertices 𝑤 ∈ 𝑊, deg 𝑤 < 𝜇𝐸, and
• 3. For any pair of vertices 𝑣, 𝑤 ∈ 𝑊, deg 𝑣, 𝑤 < 𝜀𝐸,

has a set 𝐹′ of edges with the properties a. 𝐹′ = 1 ± 𝜀′

𝜁𝑜 𝑠 ,

• b. for 𝑊′ = 𝑊 ∖ ∪𝑓∈𝐹′ 𝑓 we have 𝑊′ = 1 ± 𝜀′ 𝑜𝑓−𝜁, and

c. For all but at most 𝜀′ 𝑊′ vertices 𝑤 ∈ 𝑊′, the degree of 𝑤 in 𝐼 𝑊′ is 1 ± 𝜀′ 𝐸𝑓−𝜁 𝑠−1 .

Using the Lemma

Plan of attack

• Start with original hypergraph 𝐼0 = 𝐼 on vertex set 𝑊

0 = V

• Given a hypergraph 𝐼𝑗, apply Lemma 6.2.4 to obtain a set of edges 𝐹𝑗
• Let 𝑊

𝑗+1 = 𝑊 𝑗 ∖ ∪𝑓∈𝐹𝑗 𝑓 be the uncovered vertices

• 𝐼𝑗+1 = 𝐼 𝑊

𝑗+1 the induced hypergraph

• Once 𝑊

𝑢 is sufficiently small, cover each remaining vertex greedily

• ⇒ total size of cover is 𝑊

𝑢 + σ𝑗<𝑢 𝐹𝑗

Parameters

• With every application of the lemma, control over the distribution worsens
• Initial distribution of edges very good ⇒ lemma can be used throughout
• Work backwards to determine what is needed

Evolution of Parameters

Before applying the lemma

• 𝑜 vertices, all but 𝜀𝑜 have degree 1 ± 𝜀 𝐸
• Maximum degree < 𝜇𝐸
• Maximum codegree < 𝜀𝐸

After applying the lemma

• 1 ± 𝜀′ 𝑜𝑓−𝜁 vertices, all but 𝜀′ proportion have degree 1 ± 𝜀′ 𝐸𝑓−𝜁 𝑠−1
• Maximum degree < 𝜇𝐸, maximum codegree < 𝜀𝐸

Change of parameters

• 𝐸𝑗+1 ≔ 𝐸𝑗𝑓−𝜁 𝑠−1
• ⇒ 𝜇𝑗+1 ≔ 𝜇𝑗𝑓𝜁 𝑠−1 , 𝜀𝑗+1 ≥ 𝜀𝑗𝑓𝜁 𝑠−1
• Need 𝜀𝑗 ≤ 𝜀 𝑠, 𝜇𝑗, 𝜁, 𝜀𝑗+1 to apply lemma

Size of Vertex and Edge Sets

Vertex sets

• By Lemma 6.2.4, 𝑊

𝑗 ≤ 1 + 𝜀𝑗 𝑊 𝑗−1 𝑓−ε

• ⇒ 𝑊

𝑗 ≤ ς𝑘=1 𝑗

1 + 𝜀

𝑘

𝑜𝑓−𝑗𝜁 ≤ 1 + σ𝑘=1

𝑗

𝜀

𝑘 𝑜𝑓−𝑗𝜁

• By growing the 𝜀𝑗 fast enough, can ensure σ𝑘=1

𝑗

𝜀

𝑘 ≤ 2𝜀𝑢

Edge sets

• Lemma 6.2.4: 𝐹𝑗 ≤ 1 + 𝜀𝑗+1

𝜁 𝑊𝑗 𝑠

≤ 1 + 𝜀𝑗+1 1 + 2𝜀𝑢

𝜁𝑜𝑓−𝑗𝜁 𝑠

≤ 1 + 4𝜀𝑢

𝜁𝑜𝑓−𝑗𝜁 𝑠

Size of the Cover

Recall

• 𝑊

𝑗 ≤ 1 + 2𝜀𝑢 𝑜𝑓−𝑗𝜁 and 𝐹𝑗 ≤ 1 + 4𝜀𝑢 𝜁𝑜𝑓−𝑗𝜁 𝑠

Total size of cover

• 𝑊

𝑢 + σ𝑗=0 𝑢−1 𝐹𝑗 ≤ 1 + 2𝜀𝑢 𝑜𝑓−𝑢𝜁 + 1 + 4𝜀𝑢 𝜁𝑜 𝑠 σ𝑗=0 𝑢−1 𝑓−𝑗𝜁

≤ 1 + 4𝜀𝑢 𝑠𝑓−𝑢𝜁 +

𝜁 1−𝑓−𝜁 𝑜 𝑠

• Choosing 𝑢 large, can ensure 𝑠𝑓−𝑢𝜁 ≤ 𝜁
• 1 − 𝑓−𝜁 ≥ 1 − 1 − 𝜁 +

1 2 𝜁2

= 𝜁 1 −

1 2 𝜁

• ⇒

𝜁 1−𝑓−𝜁 ≤ 1 1−1

2𝜁 ≤ 1 + 𝜁

• ⇒ cover has size at most 1 + 4𝜀𝑢

1 + 2𝜁

𝑜 𝑠

• By choosing 𝜁, 𝜀𝑢 sufficiently small, we can ensure this is at most 1 + 𝛽

𝑜 𝑠

Piecing It Together

Proof

• Choose 𝜁, 𝜀 so that 1 + 4𝜀

1 + 2𝜁 < 1 + 𝛽, and 𝑢 so that 𝑠𝑓−𝑢𝜁 ≤ 𝜁

• Set 𝜇𝑗 ≔ 𝜆𝑓𝑗𝜁 𝑠−1 and 𝐸𝑗 ≔ 𝐸𝑓−𝑗𝜁 𝑠−1 for each 0 ≤ 𝑗 ≤ 𝑢
• Set 𝜀𝑢 ≔ 𝜀, and, for 𝑗 = 𝑢 − 1, 𝑢 − 2, … , 0, choose 𝜀𝑗 such that
• 𝜀𝑗 ≤ 𝜀 𝑠, 𝜇𝑗, 𝜁, 𝜀𝑗+1 from Lemma 6.2.4, 𝜀𝑗 ≤ 𝑓−𝜁 𝑠−1 𝜀𝑗+1 and 𝜀𝑗 ≤

1 2 𝜀𝑗+1

• Set 𝛿 ≔ 𝜀0 and 𝐸0 such that 𝐸𝑗 ≔ 𝐸0𝑓−𝑗𝜁 𝑠−1 ≥ 𝐸 𝑠, 𝜇𝑗, 𝜁, 𝜀𝑗+1 for all 𝑗
• We can then iterate the lemma 𝑢 times, giving the small cover

∎

Theorem 6.2.3 (Pippinger, 1989) For every integer 𝑠 ≥ 2 and reals 𝜆 ≥ 1 and 𝛽 > 0, there are 𝛿 = 𝛿 𝑠, 𝜆, 𝛽 > 0 and 𝐸0 = 𝐸0(𝑠, 𝜆, 𝛽) such that for every 𝑜 ≥ 𝐸 ≥ 𝐸0, any 𝑠-uniform 𝑜-vertex hypergraph 𝐼 with well-distributed edges has a cover of size at most 1 + 𝛽

𝑜 𝑠.

Any questions?

§3 The Lemma

Chapter 6: The Rödl Nibble The Probabilistic Method

Recalling the Statement

Lemma 6.2.4 For every integer 𝑠 ≥ 2 and reals 𝜇 ≥ 1, 𝜁 > 0 and 𝜀′ > 0, there are 𝜀 = 𝜀 𝑠, 𝜇, 𝜁, 𝜀′ and 𝐸0 = 𝐸0 𝑠, 𝜇, 𝜁, 𝜀′ such that, for every 𝑜 ≥ 𝐸 ≥ 𝐸0, every 𝑠-uniform 𝑜-vertex hypergraph 𝐼 = 𝑊, 𝐹 satisfying

• 1. For all vertices 𝑤 ∈ 𝑊 except at most 𝜀𝑜, deg 𝑤 = 1 ± 𝜀 𝐸,
• 2. For all vertices 𝑤 ∈ 𝑊, deg 𝑤 < 𝜇𝐸, and
• 3. For any pair of vertices 𝑣, 𝑤 ∈ 𝑊, deg 𝑣, 𝑤 < 𝜀𝐸,

has a set 𝐹′ of edges with the properties a. 𝐹′ = 1 ± 𝜀′

𝜁𝑜 𝑠 ,

• b. for 𝑊′ = 𝑊 ∖ ∪𝑓∈𝐹′ 𝑓 we have 𝑊′ = 1 ± 𝜀′ 𝑜𝑓−𝜁, and

c. For all but at most 𝜀′ 𝑊′ vertices 𝑤 ∈ 𝑊′, the degree of 𝑤 in 𝐼 𝑊′ is 1 ± 𝜀′ 𝐸𝑓−𝜁 𝑠−1 .

Proof Strategy

Selection of edges

• Select each edge to be in 𝐹′ independently at random

Analysis

• Estimate 𝐹′ , ℙ 𝑤 ∈ 𝑊′ and ℙ 𝑓 ∈ 𝐼 𝑊′
• Concentration inequalities ⇒ hypergraph statistics close to expectations
• Quantifying over vertices
• Polynomial concentration suffices
• Can use Chebyshev’s Inequality

Proof of Lemma, Part a

Proof

• Select each edge to be in 𝐹′ independently with probability 𝑞 =

𝜁 𝐸

• ⇒ 𝐹′ ∼ Bin 𝑓 𝐼 , 𝑞
• Handshake Lemma ⇒ 𝑓 𝐼 =

1 𝑠 σ𝑤 deg 𝑤

• Sum of degrees
• At least 1 − 𝜀 𝑜 ⋅ 1 − 𝜀 𝐸 + 𝜀𝑜 ⋅ 0 = 1 − 𝜀 2𝑜𝐸 ≥ 1 − 2𝜀 𝑜𝐸
• At most 1 − 𝜀 𝑜 ⋅ 1 + 𝜀 𝐸 + 𝜀𝑜 ⋅ 𝜇𝐸 = 1 − 𝜀2 + 𝜀𝜇 𝑜𝐸
• ⇒ 𝑓 𝐼 = 1 ± 𝜀1

𝑜𝐸 𝑠 for some 𝜀1 = 𝜀1 𝜀, 𝜇 → 0 as 𝜀 → 0

Lemma 6.2.4.a a. 𝐹′ = 1 ± 𝜀′

𝜁𝑜 𝑠

Proof of Lemma, Part a

Proof (cont’d)

• Recall
• Each edge selected with probability 𝑞 =

𝜁 𝐸

• 𝑓 𝐼 = 1 ± 𝜀1

𝑜𝐸 𝑠

• ⇒ 𝔽 𝐹′

= 𝑓 𝐼 𝑞 = 1 ± 𝜀1

𝜁𝑜 𝑠

• Var 𝐹′

= 𝑓 𝐼 𝑞 1 − 𝑞 ≤ 𝔽 𝐹′ = 𝑝 𝔽 𝐹′

2

• ∴ Chebyshev ⇒ with high probability, 𝐹′ = 1 ± 2𝜀1

𝜁𝑜 𝑠

∎

Lemma 6.2.4.a 𝐹′ = 1 ± 𝜀′

𝜁𝑜 𝑠 .

Proof of Lemma, Part b

Proof

• 𝑊′ = σ𝑤∈𝑊 1 𝑤∈𝑊′
• 𝔽 1 𝑤∈𝑊′

= ℙ 𝑤 ∈ 𝑊′ = 1 − 𝑞 deg 𝑤

• When deg 𝑤 = 1 ± 𝜀 𝐸:
• 𝔽 1 𝑤∈𝑊′

= 1 −

𝜁 𝐸 1±𝜀 𝐸

= 1 ± 𝜀2 𝑓−𝜁 for some 𝜀2 = 𝜀2 𝜁, 𝜀 → 0 and 𝐸 large

• At most 𝜀𝑜 exceptional vertices, for which 0 ≤ 𝔽 1 𝑤∈𝑊′

≤ 1

• ⇒ 𝔽 𝑊′

= 1 ± 𝜀3 𝑜𝑓−𝜁 for some 𝜀3 = 𝜀3 𝜁, 𝜀 → 0

Lemma 6.2.4.b For 𝑊′ = 𝑊 ∖ ∪𝑓∈𝐹′ 𝑓 we have 𝑊′ = 1 ± 𝜀′ 𝑜𝑓−𝜁.

Proof of Lemma, Part b

Proof (cont’d)

• Var 𝑊′

= σ𝑤∈𝑊 Var 1 𝑤∈𝑊′ + σ𝑣≠𝑤 Cov 1 𝑣∈𝑊′ , 1 𝑤∈𝑊′

• σ𝑤∈𝑊 Var 1 𝑤∈𝑊′

≤ σ𝑤∈𝑊 𝔽 1 𝑤∈𝑊′ = 𝔽 𝑊′

• Cov 1 𝑣∈𝑊′ , 1 𝑤∈𝑊′

= 𝔽 1 𝑣∈𝑊′ 1 𝑤∈𝑊′ − 𝔽 1 𝑣∈𝑊′ 𝔽 1 𝑤∈𝑊′ = 1 − 𝑞 deg 𝑣 +deg 𝑤 −deg 𝑣,𝑤 − 1 − 𝑞 deg 𝑣 +deg 𝑤 ≤ 1 − 𝑞 − deg 𝑣,𝑤 − 1 ≤ 1 −

𝜁 𝐸 −𝜀𝐸

− 1

• ⇒ Cov 1 𝑣∈𝑊′ , 1 𝑤∈𝑊′

≤ 𝜀4 for some 𝜀4 = 𝜀4 𝜁, 𝜀 → 0

Lemma 6.2.4.b For 𝑊′ = 𝑊 ∖ ∪𝑓∈𝐹′ 𝑓 we have 𝑊′ = 1 ± 𝜀′ 𝑜𝑓−𝜁.

Proof of Lemma, Part b

Proof (cont’d)

• Recall
• 𝔽 𝑊′

= 1 ± 𝜀3 𝑜𝑓−𝜁

• Cov 1 𝑣∈𝑊′ , 1 𝑤∈𝑊′

≤ 𝜀4

• ⇒ Var 𝑊′

≤ 𝔽 𝑊′ + 𝜀4𝑜2 ≤ 2𝜀4𝑜2

• Chebyshev: ℙ 𝑊′ ≠ 1 ± 𝜀5 𝑜𝑓−𝜁 ≤

2𝜀4𝑜2 𝜀5−𝜀3 2𝑜2𝑓−2𝜁

• This can be made arbitrarily small for appropriate choice of 𝜀5 → 0

∎

Lemma 6.2.4.b For 𝑊′ = 𝑊 ∖ ∪𝑓∈𝐹′ 𝑓 we have 𝑊′ = 1 ± 𝜀′ 𝑜𝑓−𝜁.

Proof of Lemma, Part c

Proof (outline)

• Fix a vertex 𝑤 ∈ 𝑊, and condition on 𝑤 ∈ 𝑊′
• Need to study how many edges 𝑓 ∋ 𝑤 survive in 𝐼 𝑊′
• Edge 𝑓 survives if and only if 𝑣 ∈ 𝑊′ for all 𝑣 ∈ 𝑓
• We have good control over vertices of degree 1 ± 𝜀 𝐸
• ⇒ can control edges whose vertices are all of typical degree
• Call such edges good, and bad otherwise
• ⇒ can control deg 𝑤 if most edges 𝑓 ∋ 𝑤 are good
• Shall show that degree conditions ⇒ most vertices are mostly in good edges

Lemma 6.2.4.c For all but at most 𝜀′ 𝑊′ vertices 𝑤 ∈ 𝑊′, the degree of 𝑤 in 𝐼 𝑊′ is 1 ± 𝜀′ 𝐸𝑓−𝜁 𝑠−1 .

Good Edges

Proof of i.

• At most 𝜀𝑜 vertices have deg 𝑤 ≠ 1 ± 𝜀 𝐸
• ⇒ there are at most 𝜀𝑜 ⋅ 𝜇𝐸 bad edges.
• ⇒ at most

𝜀𝜇𝑜𝐸 𝜀6𝐸 vertices can be in more than 𝜀6𝐸 bad edges

• For a suitable choice of 𝜀6 → 0, this is less than 𝜀6 − 𝜀 𝑜

∎

Claim 6.3.1 There is some 𝜀6 → 0 such that: i. all but at most 𝜀6𝑜 vertices have deg 𝑤 = 1 ± 𝜀6 𝐸, and are in at most 𝜀6𝐸 bad edges. ii. if an edge 𝑓 is good, then given some 𝑤 ∈ 𝑓, we have 𝑔 ∈ 𝐹: 𝑤 ∉ 𝑔, 𝑔 ∩ 𝑓 ≠ ∅ = 1 ± 𝜀6 𝑠 − 1 𝐸.

Good Edges

Proof of ii.

• 𝑓 good ⇒ for the 𝑠 − 1 vertices 𝑣 ∈ 𝑓, 𝑣 ≠ 𝑤, we have deg 𝑣 = 1 ± 𝜀 𝐸
• ⇒

𝑔 ∈ 𝐹: 𝑤 ∉ 𝑔, 𝑔 ∩ 𝑓 ≠ ∅ ≤ 1 + 𝜀 𝑠 − 1 𝐸

• Overcounted: edges 𝑔 that meet two vertices of 𝑓
• Co-degree bound ⇒ at most 𝑠

2 𝜀𝐸 such edges

• ⇒

𝑔 ∈ 𝐹: 𝑤 ∉ 𝑔, 𝑔 ∩ 𝑓 ≠ ∅ ≥ 1 − 𝜀 𝑠 − 1 𝐸 −

𝑠 2 𝜀𝐸

∎

Claim 6.3.1 There is some 𝜀6 → 0 such that: i. all but at most 𝜀6𝑜 vertices have deg 𝑤 = 1 ± 𝜀6 𝐸, and are in at most 𝜀6𝐸 bad edges. ii. if an edge 𝑓 is good, then given some 𝑤 ∈ 𝑓, we have 𝑔 ∈ 𝐹: 𝑤 ∉ 𝑔, 𝑔 ∩ 𝑓 ≠ ∅ = 1 ± 𝜀6 𝑠 − 1 𝐸.

Survival of Good Edges

Proof

• 𝑤 ∈ 𝑊′ ⇒ no edge containing 𝑤 was selected in 𝐹′
• 𝑓 ⊆ 𝑊′ ⇒ every 𝑣 ∈ 𝑓 is also in 𝑊′
• ⇒ no edge 𝑔 ∈ 𝐹 with 𝑔 ∩ 𝑓 ≠ ∅ is selected in 𝐹′
• By assumption, this is true for every 𝑔 ∋ 𝑤
• ⇒ need only consider 𝑔 ∈ 𝐹: 𝑤 ∉ 𝑔, 𝑔 ∩ 𝑓 ≠ ∅
• Claim 6.3.1.ii ⇒ there are 1 ± 𝜀6

𝑠 − 1 𝐸 such edges

• Probability none are selected in 𝐹′ is 1 − 𝑞

1±𝜀6 𝑠−1 𝐸

• 𝑞 =

𝜁 𝐸 ⇒ this is 1 ± 𝜀7 𝑓−𝜁 𝑠−1

∎

Claim 6.3.2 There is some 𝜀7 → 0 such that, if we condition on 𝑤 ∈ 𝑊′, and 𝑓 is a good edge containing 𝑤, then ℙ 𝑓 ⊆ 𝑊′ = 1 ± 𝜀7 𝑓−𝜁 𝑠−1 .

Expected Degrees

Proof

• For each edge 𝑓 ∈ 𝐹, let 1𝑓 be the indicator for the event 𝑓 ⊆ 𝑊′
• ⇒ degree of 𝑤 in 𝐼 𝑊′ is σ𝑓∋𝑤 1𝑓
• At most 𝜀6𝐸 bad edges containing 𝑤
• deg′ 𝑤 = σ𝑓∋𝑤,𝑓 good 1𝑓 ± 𝜀6𝐸
• Number of good edges containing 𝑤 is 1 ± 𝜀 ± 𝜀6 𝐸
• Claim 6.3.2 ⇒ 𝔽 1𝑓 = 1 ± 𝜀7 𝑓−𝜁 𝑠−1 for every good 𝑓 ∋ 𝑤
• ⇒ 𝔽 deg′ 𝑤

= 1 ± 𝜀 ± 𝜀6 1 ± 𝜀7 𝐸𝑓−𝜁 𝑠−1 ± 𝜀6𝐸 ∎

Claim 6.3.3 There is some 𝜀8 → 0 such that, if 𝑤 is a vertex as in Claim 6.3.1.i and we condition on 𝑤 ∈ 𝑊′, then the expected degree deg′ 𝑤 of 𝑤 in 𝐼 𝑊′ is 1 ± 𝜀8 𝐸𝑓−𝜁 𝑠−1 .

Variance in Degrees

Proof

• As usual, Var deg′ 𝑤

≤ 𝔽 deg′ 𝑤 + σ𝑤∈𝑓,𝑔;𝑓≠𝑔 Cov 1𝑓, 1𝑔

• Contribution to sum from bad edges is at most 𝜀6 1 + 𝜀 𝐸2
• Fix good 𝑓 ∋ 𝑤, and estimate σ𝑔 good:𝑤∈𝑔≠𝑓 Cov 1𝑓, 1𝑔
• Let 𝑈 𝑓 = ℎ ∈ 𝐹: 𝑤 ∉ ℎ, 𝑤 ∩ 𝑓 ≠ ∅ , and let 𝑢 𝑓, 𝑔 = 𝑈 𝑓 ∩ 𝑈 𝑔
• Cov 1𝑓, 1𝑔 = 𝔽 1𝑓1𝑔 − 𝔽 1𝑓 𝔽 1𝑔

= 1 − 𝑞

𝑈 𝑓 ∪𝑈 𝑔

− 1 − 𝑞

𝑈 𝑓 + 𝑈 𝑔

≤ 1 − 𝑞 −𝑢 𝑓,𝑔 − 1

Claim 6.3.4 There is some 𝜀9 → 0 such that, if 𝑤 is a vertex as in Claim 6.3.1.i and we condition on 𝑤 ∈ 𝑊′, then Var deg′ 𝑤 ≤ 𝜀9𝐸2.

Variance in Degrees

Proof (cont’d)

• Recall
• 𝑈 𝑓 = ℎ ∈ 𝐹: 𝑤 ∉ ℎ, 𝑤 ∩ 𝑓 ≠ ∅ and 𝑢 𝑓, 𝑔 = 𝑈 𝑓 ∩ 𝑈 𝑔
• Cov 1𝑓, 1𝑔 ≤ 1 − 𝑞 −𝑢 𝑓,𝑔 − 1
• For each 𝑣 ∈ 𝑓, deg 𝑣, 𝑤 ≤ 𝜀𝐸
• ⇒ at most 𝑠 − 1 𝜀𝐸 edges 𝑔 with 𝑓 ∩ 𝑔 ≥ 2
• Otherwise 𝑓 ∩ 𝑔 = 𝑤
• ⇒ for all ℎ ∈ 𝑈 𝑓 ∩ 𝑈 𝑔 there are 𝑣 ∈ 𝑓 ∖ 𝑤 , 𝑣′ ∈ 𝑔 ∖ 𝑤 with 𝑣, 𝑣′ ∈ ℎ
• ⇒ 𝑢 𝑓, 𝑔 ≤ 𝑠 − 1 2𝜀𝐸
• ⇒ Cov 1𝑓, 1𝑔 ≤ 1 −

𝜁 𝐸 − 𝑠−1 2𝜀𝐸

− 1 ≤ 𝑠2𝜁𝜀

Claim 6.3.4 There is some 𝜀9 → 0 such that, if 𝑤 is a vertex as in Claim 6.3.1.i and we condition on 𝑤 ∈ 𝑊′, then Var deg′ 𝑤 ≤ 𝜀9𝐸2.

Variance in Degrees

Proof (cont’d)

• Var deg′ 𝑤

≤ 2𝜀6𝐸2 + σ𝑤∈𝑓,𝑓 good σ𝑤∈𝑔,𝑔 good Cov 1𝑓, 1𝑔 ≤ 2𝜀6𝐸2 + σ𝑤∈𝑓,𝑓 good 𝑠 − 1 𝜀𝐸 + σ𝑔 good,𝑔∩𝑓= 𝑤 Cov 1𝑓, 1𝑔 ≤ 2𝜀6𝐸2 + σ𝑤∈𝑓,𝑓 good 𝑠 − 1 𝜀𝐸 + σ𝑔 good,𝑔∩𝑓= 𝑤 𝑠2𝜁𝜀 ≤ 2𝜀6𝐸2 + 1 + 𝜀 𝐸 ⋅ 𝑠 − 1 𝜀𝐸 + 1 + 𝜀 𝐸 ⋅ 𝑠2𝜁𝜀

• For appropriate 𝜀9 → 0, this is at most 𝜀9𝐸2

∎

Claim 6.3.4 There is some 𝜀9 → 0 such that, if 𝑤 is a vertex as in Claim 6.3.1.i and we condition on 𝑤 ∈ 𝑊′, then Var deg′ 𝑤 ≤ 𝜀9𝐸2.

Completing the Proof

Proof

• All but at most 𝜀6𝑜 vertices are as in Claim 6.3.1.i; can ignore the rest
• For such a vertex 𝑤, conditioning on 𝑤 ∈ 𝑊′:
• Claim 6.3.3: 𝔽 deg′ 𝑤

= 1 ± 𝜀8 𝐸𝑓−𝜁 𝑠−1

• Claim 6.3.4: Var deg′ 𝑤

≤ 𝜀9𝐸2

• Chebyshev: for some 𝜀10 → 0, ℙ deg 𝑤 ≠ 1 ± 𝜀10 𝐸𝑓−𝜁 𝑠−1

≤ 𝜀6

• Markov: the probability of having more than 2𝜀6 𝑊′ such vertices in 𝑊′

whose degree is not 1 ± 𝜀10 𝐸𝑓−𝜁 𝑠−1 is less than

1 2

∎

Lemma 6.2.4.c For all but at most 𝜀′ 𝑊′ vertices 𝑤 ∈ 𝑊′, the degree of 𝑤 in 𝐼 𝑊′ is 1 ± 𝜀′ 𝐸𝑓−𝜁 𝑠−1 .

Epilogue

Central question

• For which 𝑜 does a 𝑢-(𝑜, 𝑙, 1) design exist?
• Divisibility conditions ⇒ infinite sequence of possible values
• These conditions are necessary, but not sufficient

Erdős-Hanani Conjecture / Rödl’s Theorem

• ⇒ for all large 𝑜, asymptotic designs exist

Exact results

• Wilson (1972-1975): 𝑢 = 2, 𝑙 ≥ 3, 𝑜 large and satisfying divisibility conditions
• Keevash (2014+): generalised Wilson to all 𝑢
• Follows the steps of Rödl Nibble, but uses an algebraic construction to complete design
• Glock, Kühn, Lo and Osthus (2016+): new proof of existence of designs
• Proof is purely combinatorial/probabilistic

Any questions?