Upper density of monochromatic infinite paths

Corsten, Jan; DeBiasio, Louis; Lamaison, Ander; Lang, Richard

doi:10.19086/aic.10810

Cited by 9 publications

(8 citation statements)

References 7 publications

(15 reference statements)

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“…Progress towards this conjecture was made by Lo, Sanhueza-Matamala and Wang [19], who raised the lower bound to (9 + √ 17)/16 ≈ 0.82019. Corsten, DeBiasio, Lamaison and Lang [7] finally proved that Rd(P ∞ ) = (12 + √ 8)/17 ≈ 0.87226, thereby settling the problem for two colors. In this paper, we initiate a systematic study of Ramsey densities for other infinite graphs.…”

Section: Introductionmentioning

confidence: 97%

Density of monochromatic infinite subgraphs II

Corsten¹,

DeBiasio²,

McKenney³

2020

Preprint

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In 1967, Gerencsér and Gyárfás proved the following seminal result in graph-Ramsey theory: every 2-colored K n contains a monochromatic path on (2n + 1)/3 vertices, and this is best possible. In 1993, Erdős and Galvin started the investigation of the analogous problem in countably infinite graphs. After a series of improvements, this problem was recently settled: in every 2-coloring of K N there is a monochromatic infinite path with upper density at least (12 + √ 8)/17, and there exists a 2-coloring which shows this is best possible. Since 1967, there have been hundreds of papers on finite graph-Ramsey theory with many of the most important results being motivated by a series of conjectures of Burr and Erdős about graphs with linear Ramsey numbers. In a sense, this paper begins a systematic study of infinite graph-Ramsey theory, focusing on infinite analogues of these conjectures. The following are some of our main results.(i) Let G be a countably infinite, (one-way) locally finite graph with chromatic number χ < ∞. Every 2-colored K N contains a monochromatic copy of G with upper density at least 1 2(χ−1) . (ii) Let G be a countably infinite graph having the property that there exists a finite set X ⊆ V (G) such that G − X has no finite dominating set (in particular, graphs with bounded degeneracy have this property, as does the infinite random graph). Every finitely-colored K N contains a monochromatic copy of G with positive upper density. (iii) Let T be a countably infinite tree. Every 2-colored K N contains a monochromatic copy of T of upper density at least 1/2. In particular, this is best possible for T ∞ , the tree in which every vertex has infinite degree. (iv) Surprisingly, there exist connected graphs G such that every 2-colored K N contains a monochromatic copy of G which covers all but finitely many vertices of N. In fact, we classify all forests with this property.

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Section: Introductionmentioning

confidence: 97%

Density of monochromatic infinite subgraphs II

Corsten¹,

DeBiasio²,

McKenney³

2020

Preprint

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“…This implies that T i ≤ 0 for some i (see for example [1]). This proves that S ≥ 2γ 2 +2γ+8+ √ 32(1−γ) (γ+1) 3 , which produces the desired value of f (λ). Proposition 4.…”

Section: A Proof Of Lemmamentioning

confidence: 65%

“…They proved that 2/3 ≤ ρ(P ∞ ) ≤ 8/9. After some improvements on these bounds in [4,8], the exact value of P ∞ was determined by Corsten, DeBiasio, Lang and the author [3] as ρ(P ∞ ) = (12 + √ 8)/17 ≈ 0.87226. The parameter ρ(H) for general H was first introduced by DeBiasio and McKenney [4].…”

Section: Introductionmentioning

confidence: 99%

Ramsey upper density of infinite graphs

Lamaison¹

2020

Preprint

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For a fixed infinite graph H, we study the largest density of a monochromatic subgraph isomorphic to H that can be found in every two-coloring of the edges of K N . This is called the Ramsey upper density of H, and was introduced by Erdős and Galvin. Recently [3], the Ramsey upper density of the infinite path was determined. Here, we find the value of this density for all locally finite graphs H up to a factor of 2, answering a question of DeBiasio and McKenney.We also find the exact density for a wide class of bipartite graphs, including all locally finite forests. Our approach relates this problem to the solution of an optimization problem for continuous functions. We show that, under certain conditions, the density depends only on the chromatic number of H, the number of components of H, and the expansion ratio |N (I)|/|I| of the independent sets of H.

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“…After the submission of this paper, we learned that Corsten, DeBiasio, Lamaison and Lang [2] have obtained an improved version of Theorem 1.…”

Section: Remarkmentioning

confidence: 99%

Density of Monochromatic Infinite Paths

Sanhueza‐Matamala

Wang

2018

Electron. J. Combin.

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For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.

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Upper density of monochromatic infinite paths

Abstract: We prove that in every 2-colouring of the edges of K N there exists a monochromatic infinite path P such that V (P ) has upper density at least (12 + √ 8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin.

Cited by 9 publications

References 7 publications

Density of monochromatic infinite subgraphs II

Density of monochromatic infinite subgraphs II

Ramsey upper density of infinite graphs

Density of Monochromatic Infinite Paths

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