Occupancy fraction, fractional colouring, and triangle fraction

Given $\varepsilon>0$, there exists $f_0$ such that, if $f_0 \le f \le \Delta^2+1$, then for any graph $G$ on $n$ vertices of maximum degree $\Delta$ in which the neighbourhood of every vertex in $G$ spans at most $\Delta^2/f$ edges, (i) an independent set of $G$ drawn uniformly at random has at least $(1/2-\varepsilon)(n/\Delta)\log f$ vertices in expectation, and (ii) the fractional chromatic number of $G$ is at most $(2+\varepsilon)\Delta/\log f$. These bounds cannot in general be improved by more than a factor $2$ asymptotically. One may view these as stronger versions of results of Ajtai, Koml\'os and Szemer\'edi (1981) and Shearer (1983). The proofs use a tight analysis of the hard-core model.


Introduction
By a deletion argument, Ajtai, Komlós and Szemerédi [2] noted a more general statement as corollary to their seminal bound on the independence number of triangle-free graphs. There is some C > 0 and some f 0 such that, in any graph on n vertices of maximum degree ∆ with at most ∆ 2 n/f triangles, where f 0 < f < ∆, there is an independent set of size at least C(n/∆) log f . In other words, an upper bound on the fraction of triangles in the graph yields a corresponding lower bound on independence number. Somewhat later, Alon, Krivelevich and Sudakov [3] proved a stronger version of this in terms of an upper bound on the chromatic number. Recently, using a sophisticated "stochastic local search" framework, Achlioptas, Iliopoulos and Sinclair [1] tightened the result of [3], corresponding to a constant C above of around 1/4 in general 1 . In fact, shortly after the work in [2], using a sharper bootstrapping from the triangle-free case, Shearer [12] had improved the above statement on independence number essentially 2 as follows.
The case f = ∆ 2−o(1) as ∆ → ∞ includes the triangle-free case and yields the best to date asymptotic lower bound on the off-diagonal Ramsey numbers. The asymptotic factor 1/2 cannot be improved above 1, due to random regular graphs; see Section 5 for more details on sharpness.
Our main contribution is to give two stronger forms of Theorem 1, one on occupancy fraction (see Theorem 5 below), the other on fractional chromatic number, combining for the result promised in the abstract. We show how either easily implies Theorem 1 in Section 2.
then for any graph G on n vertices of maximum degree ∆ in which the neighbourhood of every vertex in G spans at most ∆ 2 /f edges, (i) an independent set of G drawn uniformly at random has at least (1/2 − ε)(n/∆) log f vertices in expectation, and (ii) the fractional chromatic number of G is at most (2 + ε)∆/ log f . We prove Theorem 2 by an analysis of the hard-core model. In Section 5, we give some indication that our application of this analysis is essentially tight. The same method was used for similar results specific to triangle-free graphs [8,7]; to an extent, the present work generalises that earlier work.
Theorem 2(ii) and the results in [1] hint at their common strengthening.
Conjecture 3. Given ε > 0, there exists f 0 such that, if f 0 ≤ f ≤ ∆ 2 + 1, then any graph of maximum degree ∆ in which the neighbourhood of every vertex spans at most ∆ 2 /f edges has (list) chromatic number at most (2 + ε)∆/ log f .
In Section 6, motivated by quantitative Ramsey theory, we briefly discuss a more basic problem setting in terms of bounded triangle fraction.
1.1. Notation and preliminaries. We write I(G) for the set of independent sets (including the empty set) of a graph G. Given λ > 0, the hard-core model on G at fugacity λ is a probability distribution on I(G), where each I ∈ I(G) occurs with probability proportional to λ |I| . Writing I for the random independent set, we have where the normalising term in the denominator is the partition function (or independence polynomial) Z G (λ) = I∈I(G) λ |I| . The occupancy fraction is E|I|/|V (G)|. Note that this is a lower bound on the proportion of vertices in a largest independent set of G.
If µ denotes the standard Lesbegue measure on R, then a fractional colouring of G is an assignment w : I(G) → 2 R of pairwise disjoint measurable subsets of R to independent sets such that I∈I(G),I∋v µ(w(I)) ≥ 1 for all v ∈ V (G). The total weight of the fractional colouring is I∈I(G) µ(w(I)). The fractional chromatic number χ f (G) of G is the minimum total weight in a fractional colouring of G. Note that the reciprocal is a lower bound on the proportion of vertices in a largest independent set of G.
We make use of a special case of a lemma from our earlier work [7].
, cf. also [11]). Let G be a graph of maximum degree ∆, and α, β > 0 be positive reals. Suppose that for every induced subgraph H ⊆ G, there is a probability distribution on I(H) such that, writing I H for the random independent set from this distribution, for each v ∈ V (H) we have Then

The main result
We next discuss our main result, Theorem 2, in slightly deeper context. We in fact show a sharp, general lower bound on occupancy fraction for graphs of bounded local triangle fraction, to which Theorem 2(i) is corollary.
Theorem 5. Suppose f, ∆, λ satisfy that f ≤ ∆ 2 + 1 and, as f → ∞, that In any graph G of maximum degree ∆ in which the neighbourhood of every vertex spans at most ∆ 2 /f edges, writing I for the random independent set from the hard-core model on G at fugacity λ, the occupancy fraction satisfies This may be viewed as generalising [8,Thm. 3]. By monotonicity of the occupancy fraction in λ (see e.g. [8, Prop. 1]), and the fact that a uniform choice from I(G) is a hard-core distribution with λ = 1, Theorem 2(i) follows from Theorem 5 with λ = f 1/(2+ε/2) /∆. Theorem 5 is asymptotically optimal. More specifically, in [8] it was shown how the analysis of [4] yields that, for any fugacity λ = o(1) in the range allowed in Theorem 5, the random ∆-regular graph (conditioned to be triangle-free) with high probability has occupancy fraction asymptotically equal to the bound in Theorem 5. In Section 5, we show our methods break down for λ outside this range, so that new ideas are needed for any improvement in the bound for larger λ.
Moreover, the asymptotic bounds of Theorems 1 and 2 cannot be improved, for any valid choice of f as a function of ∆, by more than a factor of between 2 and 4. This limits the hypothetical range of λ in Theorem 5. This follows by considering largest independent sets in a random regular construction or in a suitable blow-up of that construction [12]; see Section 5.
Observe that Theorem 2(ii) trivially fails with a global, rather than local, triangle fraction condition by the presence of a (∆ + 1)-vertex clique as a subgraph. So Theorems 1 and 2 may appear incompatible, since the former has a global condition, while the latter has a local one. Nevertheless, either assertion in Theorem 2 is indeed (strictly) stronger.
Proof of Theorem 1. Without loss of generality we may assume that ε > 0 is small enough so that ( Then H is a graph of maximum degree ∆ on at least (1 − 3ε 2 )n vertices such that the neighbourhood of any vertex spans at most ∆ 2 /(ε 2 f ) edges. Provided we take f large enough, either of (i) and (ii) in Theorem 2 implies that H, and thus G, contains an independent set of size where on the first line we used that ε 2 f ≥ f 1−ε 2 for f large enough.
3. An analysis of the hard-core model A crucial ingredient in the proofs is an occupancy guarantee from the hard-core model, which we establish in Lemma 7 below. This refines an analysis given in [8]. Given G, I ∈ I(G), and v ∈ V (G), let us call a neighbour u ∈ N (v) of v externally uncovered by I if u / ∈ N (I \ N (v)).
Lemma 6. Let G be a graph and λ > 0. Let I be an independent set drawn from the hard-core model at fugacity λ on G.
(i) For every v ∈ V (G), writing F v for the subgraph of G induced by the neighbours of v externally uncovered by I, (ii) Moreover, Proof. The first part follows from two applications of the spatial Markov property of the hard-core model. First, we have because conditioned on a value I \ {v} = J such that J ∩ N (v) = ∅ there are two realisations of I, namely J and J ∪ {v}, giving and conditioned on I \ {v} = J such that J ∩ N (v) = ∅, v cannot be in I. Second, the spatial Markov property gives that I ∩ N (v) is a random independent set drawn from the hard-core model on F v . Then I ∩ N (v) = ∅ if and only if this random independent set in F v is empty. It follows that since the graph on |V (F v )| vertices with largest partition function is the graph with no edges, and by convexity. This completes the proof of (i).
Then (ii) follows by convexity: Lemma 7. Let G be a graph of maximum degree ∆ in which the neighbourhood of every vertex in G spans at most ∆ 2 /f edges and λ, α, β > 0. Let I be an independent set drawn from the hard-core model at fugacity λ on G.
Then we have, for every v ∈ V (G), Proof. Write F v for the graph induced by the neighbours of v externally uncovered by I and z v = E|V (F v )|. By Lemma 6(i) we have For the other term, we apply Lemma 6(ii) to the graph F v , for which by If J is an independent set drawn from the hard-core model at fugacity λ on F v , then by convexity so (1) follows. For (2), by above we may bound E|I| in two distinct ways: where z is the expected number of externally uncovered neighbours of a uniformly random vertex. Now (2) follows.
Proof of Theorem 2(ii). Supposing that we have chosen α, β, and λ, write By (1) in Lemma 7 and Lemma 4, we have χ f (G) ≤ α + β∆, provided g(x) ≥ 1 for all x ≥ 0. It is easy to verify that with α, β, λ > 0 the function g is strictly convex, so the minimum of g(x) occurs when g ′ (x) = 0, or As before, let z satisfy (3). Then by choosing the minimum of g occurs at z. Now the equations g(z) = 1 and (5) give us values of α and β in terms of λ, ∆, and f . Using (3), this means and hence by Lemma 4 we obtain We take λ = o(1) as f → ∞ given by ∆ log(1 + λ) = f 1/(2+ε/2) , and note the analysis of (3) gives in this case that z log(1 + λ) = W (∆λ) + o (1). Thus provided f 0 is taken large enough.

Occupancy fraction.
Since the occupancy fraction is increasing in λ, it might be intuitive that the lower bound on occupancy fraction that results from the proof of Theorem 5 is also increasing. This is true only up to a point, just as in [8]. Already for λ slightly larger than admissible for Theorem 5, under mild assumptions, the method breaks down in the sense that the resulting lower bound is asymptotically smaller. In this case a novel analysis would be necessary; there is almost no slack in our treatment of (3).
The choice of λ used to obtain Theorem 2 is just shy of the above range. Substituting the extremes of the interval (4) into the second argument of (2), we derive the following two expressions as f → ∞, the larger of which necessarily bounds the best guarantee to expect from our approach. First, Second, using the properties of W and the assumed bounds on λ,

5.2.
Independence number and fractional chromatic number. By an analysis of the random regular graph [10], there are triangle-free ∆-regular graphs G ∆ in which as ∆ → ∞ the largest independent set has size at most So G ∆ certifies Theorems 1 and 2 to be sharp up to an asymptotic factor 2 provided f = ∆ 2−o(1) as ∆ → ∞. For smaller f , let us for completeness reiterate an observation from [12]. For f < ∆/2, let d = f − 1 and let bG d be the graph obtained from G d by substituting each vertex with a clique of size b = ⌊∆/f ⌋. Then bG d is regular of degree bf − 1 ≤ ∆ such that each neighbourhood contains at most b 2 f /2 ≤ ∆ 2 /(2f ) edges, and so bG d has at most ∆ 2 |V (bG d )|/f triangles. In G d the largest independent set is of size The same is clearly true in bG d , and this is an asymptotic factor 4 greater than the lower bound in Theorem 1. Last, observe that for f ≥ ∆/2 and f ≤ ∆ 2−Ω(1) , G ∆ certifies that Theorems 1 and 2 are at most an asymptotic factor 4 from extremal, and so this holds throughout the range of f .

A more basic setting
Due to the close link with off-diagonal Ramsey numbers, we wonder, what occurs when we drop the degree bounding parameter? This may be variously interpreted. For example, over graphs on n vertices with at most n 3 /f triangles, what is the best lower bound on the independence number?
One can deduce nearly the correct answer to an alternative, local version of this question: over graphs on n vertices such that each vertex v is contained in at most deg(v) 2 /f triangles, where 2 ≤ f ≤ (n − 1) 2 + 1, what is the best lower bound on the independence number? By a comparison of Theorem 1 and Turán's theorem (applied to a largest neighbourhood), as f → ∞ there must be an independent of size at least Shearer's bound on off-diagonal Ramsey numbers to cover any f ≥ n 1−o (1) .
Over the range of f as a function of n, (6) is asymptotically sharp up to some reasonably small constant factor by considering the final output of the triangle-free process [5,9] or a blow-up of that graph by cliques. We remark that the same argument as above yields a similar bound as in (6) for a more local version, where f = deg(v) a +1 for some fixed a ∈ [0, 2].
Is this the correct asymptotic order for the largest list chromatic number? Is the extra factor 2 unnecessary for the chromatic number? Even improving the bound only for the fractional chromatic number by a factor 2 would be very interesting. This generalises recent conjectures of Cames van Batenburg and a subset of the authors [6], for the triangle-free case f = (n − 1) 2 + 1.