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Without a large $ϵ$ -sparse induced subgraph, the number of $k$ -vertex stable sets is bounded

Statement

Let $ϵ \in (0, 1]$ , let $0 \leq ℓ \leq k$ be integers, and let $u, n$ be positive integers with

(1 - ϵ)^{ℓ} n \leq u .

Let $G$ be a finite graph on $n$ vertices such that every subset $S \subseteq V (G)$ with $∣ S ∣ \geq u$ induces a graph $G [S]$ of maximum degree at least $ϵ ∣ S ∣ - 1$ . Then $G$ has at most

(\binom{n}{ℓ}) (\binom{u}{k - ℓ})

stable sets of size $k$ .

In particular, the same bound holds whenever $G$ has no $ϵ$ -sparse induced subgraph on $u$ or more vertices.

Facts & Assumptions

Given: A real $ϵ \in (0, 1]$ , integers $0 \leq ℓ \leq k$ , positive integers $u, n$ with $(1 - ϵ)^{ℓ} n \leq u$ , and an $n$ -vertex graph $G$ satisfying the maximum-degree hypothesis in the Statement.

[L1]

Stable sets are vertex sets with no adjacent pair (Cliques, stable sets, the clique number $ω (G)$ and stability number $α (G)$ ).

[L2]

Binomial coefficients count subsets, and Pascal's rule is $(\binom{n}{ℓ - 1}) + (\binom{n}{ℓ}) = (\binom{n + 1}{ℓ})$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ , Pascal's rule $(\binom{n + 1}{k + 1}) = (\binom{n}{k}) + (\binom{n}{k + 1})$ , and the hockey-stick identity $\sum_{i \leq n} (\binom{i}{k}) = (\binom{n + 1}{k + 1})$ ).

[L3]

If a vertex set $S$ induces a graph whose maximum degree is less than $ϵ ∣ S ∣$ , then $S$ is $ϵ$ -sparse; equivalently, the failure of $ϵ$ -sparsity forces some vertex degree to exceed $ϵ ∣ S ∣$ ( $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets, A set is $c$ -sparse exactly when the maximum degree of the graph it induces is at most $c$ times its size).

Proof

technique · induction

1.1

[base] If $ℓ = 0$ , then the hypothesis $(1 - ϵ)^{ℓ} n \leq u$ reads $n \leq u$ . Every stable $k$ -set is a $k$ -element subset of the $n$ -vertex set, so there are at most $(\binom{n}{k}) \leq (\binom{u}{k}) = (\binom{n}{0}) (\binom{u}{k})$ of them by [L2].

1.2

[ih] Assume $ℓ \geq 1$ and that the claim holds for every admissible parameter tuple with smaller value of $ℓ + n$ .

base

1.3

If $u \geq n$ , then every stable $k$ -set is a $k$ -element subset of the $n$ -vertex set, so there are at most $(\binom{n}{k}) \leq (\binom{n}{ℓ}) (\binom{n - ℓ}{k - ℓ}) \leq (\binom{n}{ℓ}) (\binom{u}{k - ℓ})$ of them by [L2]. Thus the claim is immediate in this case. We may therefore assume $u < n$ . Take $v \in V (G)$ of maximum degree. Applying the hypothesis to $S = V (G)$ gives $\deg_{G} (v) \geq ϵ n - 1$ . Let $U : = V (G) ∖ ({v} \cup N_{G} (v))$ , so $∣ U ∣ \leq (1 - ϵ) n$ .

L2givenchoosealgebra

2.1

Stable $k$ -sets containing $v$ correspond exactly to stable $(k - 1)$ -sets of $G [U]$ . Since $(1 - ϵ)^{ℓ - 1} ∣ U ∣ \leq (1 - ϵ)^{ℓ} n \leq u$ , the induction hypothesis applied to $G [U]$ with parameters $(ℓ - 1, k - 1)$ shows that there are at most $(\binom{∣ U ∣}{ℓ - 1}) (\binom{u}{k - ℓ}) \leq (\binom{n - 1}{ℓ - 1}) (\binom{u}{k - ℓ})$ such stable sets.

step 1.2step 1.3L2

2.2

Stable $k$ -sets avoiding $v$ are stable $k$ -sets of $G - v$ . Any subset of $V (G - v)$ with at least $u$ vertices is also a subset of $V (G)$ , so it still satisfies the maximum-degree hypothesis. The induction hypothesis applied to $G - v$ with parameters $(ℓ, k)$ therefore bounds their number by $(\binom{n - 1}{ℓ}) (\binom{u}{k - ℓ})$ .

step 1.2step 1.3

3.1

Adding the bounds from steps 2.1 and 2.2 and using Pascal's rule from [L2] gives at most $((\binom{n - 1}{ℓ - 1}) + (\binom{n - 1}{ℓ})) (\binom{u}{k - ℓ}) = (\binom{n}{ℓ}) (\binom{u}{k - ℓ})$ stable $k$ -sets in $G$ , in the sense of [L1]. If $G$ has no $ϵ$ -sparse induced subgraph on $u$ or more vertices, then [L3] shows that every such induced subgraph has a vertex of degree exceeding $ϵ ∣ S ∣$ , hence in particular at least $ϵ ∣ S ∣ - 1$ , so the same bound applies in that situation as well.

step 2.1step 2.2L1L2L3discharge-induction∎

Depends on

Used by

Every finite family with the Erdős–Hajnal property is viral Theorem

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Bucić, J. Fox, and H. T. Pham, Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures, Lemma 12 (standard reference, not scraped)
T. H. Nguyen, Notes on Recent Work on the Erdős–Hajnal Conjecture, Lemma 1.5 (standard reference, not scraped)

Without a large ϵϵ-sparse induced subgraph, the number of kk-vertex stable sets is bounded