Counterexamples to a conjecture of Harris on Hall ratio

Blumenthal, Adam; Lidický, Bernard; Martin, Ryan; Norin, Sergey; Pfender, Florian; Volec, Jan

doi:10.48550/arxiv.1811.11116

Cited by 1 publication

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“…For the second, it was only very recently shown by Dvořák, Ossona de Mendez, and Wu [23] (cf. also [12]) the existence of a family of graphs with Hall ratios at most 18 and unbounded fractional chromatic numbers. It remains an interesting open problem of Harris [32] to determine whether ρ(G) is always at least some constant fraction of χ f (G) for triangle-free graphs G. For the third inequality in (5), it is well known that the Kneser graphs [44].…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Graph structure via local occupancy

Davies¹,

Kang²,

Pirot³

et al. 2020

Preprint

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The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymptotics due to Shearer (1983). The present work strengthens this in two ways: first, by guaranteeing stronger graph structure in terms of colourings through applications of the Lovász local lemma; and second, by extending beyond triangle-free graphs in terms of local sparsity, treating for example graphs of bounded local edge density, of bounded local Hall ratio, and of bounded clique number. This generalises and improves upon much other earlier work, including that of Shearer (1995), Alon (1996 and Alon, Krivelevich and Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common framework built around the hard-core model. It pivots on a property we call local occupancy, giving a clean separation between the methods for deriving graph structure with probabilistic information and verifying the requisite probabilistic information itself.

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