Density of monochromatic infinite subgraphs II

TL;DR

The paper develops a systematic, density-centric framework for infinite graph Ramsey theory, introducing Ramsey upper density and the notions of Ramsey-dense graphs and ruling numbers. It establishes lower bounds for Rd_r(G) for one-way locally finite graphs with finite chromatic number and finite maximum degree, and proves key embedding results via ultrafilters to obtain monochromatic copies with positive density. It demonstrates strong density properties for trees (notably Rd(T) ≥ 1/2 and Rd(T_∞) = 1/2) and classifies cofinitely embeddable forests, linking degeneracy, ruling number, and cofiniteness. The work extends to bipartite graphs, coideals, and infinite-ruled graphs, offering a broad, interconnected landscape of infinite Ramsey densities and several open problems that bridge finite- and infinite-graph Ramsey theory with potential applications to SEO contexts and first-pass analyses.

Abstract

In 1967, Gerencsér and Gyárfás proved a result which is considered the starting point of graph-Ramsey theory: In every 2-coloring of $K_n$ there is a monochromatic path on $\lceil(2n+1)/3\rceil$ vertices, and this is best possible. There have since been hundreds of papers on graph-Ramsey theory with some of the most important results being motivated by a series of conjectures of Burr and Erd\H os regarding the Ramsey numbers of trees, graphs with bounded maximum degree, and graphs with bounded degeneracy. In 1993, Erd\H os and Galvin \cite{EG} began the investigation of a countably infinite analogue of the Gerencsér and Gyárfás result: What is the largest $d$ such that in every $2$-coloring of $K_\mathbb{N}$ there is a monochromatic infinite path with upper density at least $d$. Erd\H os and Galvin showed that $2/3\leq d\leq 8/9$, and after a series of recent improvements, it was finally shown that $d={(12+\sqrt{8})}/{17}$. This paper begins a systematic study of quantitative countably infinite graph-Ramsey theory, focusing on infinite analogues of the Burr-Erdős conjectures. We obtain some results which are analogous to what is known in finite case, and other (unexpected) results which have no analogue in the finite case.

Figures (5)

• Figure 1: The lightly shaded area represents graphs which are Ramsey-dense. The blue text represents graphs $G$ for which $\mathop{\mathrm{\overline{Rd}}}\nolimits(G)>0$. The red text represents graphs $G$ which are Ramsey-dense, but $\mathop{\mathrm{\overline{Rd}}}\nolimits(G)=0$.
• Figure 2: An example of the proof of \ref{['thm:locally-finite']}.(ii). In this example, $G$ will be embedded in blue into $W_4\cup A_{3,i_3}\cup A_{1,i_1}$ such that $W_4\subseteq V(G)$.
• Figure 3: An example of the proof of \ref{['thm:locally-finite']}.(iii) with $r=3$ and $k=3$. Here we have highlighted the sequence $A_{(1,2,1,2)},A_{(1,2,1)},A_{(1,2)},A_{(1)},A_{\emptyset}$ and note that some color, in this case red, must appear at least twice, which means we can embed $G$ into $A_{(1,2,1,2)}\cup A_{(1,2,1)}\cup A_{\emptyset}$ in such a way that $A_{(1,2,1,2)}$ is covered.
• Figure 4: An example of the coloring from \ref{['prop:selfint']} in the case when $k=2$. The shaded areas denote cliques of the respective colors and a blue/solid (red/dashed) arrow from one part to another indicates that vertices in the first part have cofinitely many blue (red) neighbors in the second part.
• Figure 5: The shaded areas denote cliques of the respective colors and a blue/solid (red/dashed) arrow from $V_i$ to $V_j$ indicates that vertices in $V_i$ have cofinitely many blue (red) neighbors in $V)j$. On the right we have an example of the relevant edges in the case where we are embedding a blue copy of $T\in \mathcal{T}^*$ with root $t$ in $V_0$.

Theorems & Definitions (99)

• Conjecture 1.1: Burr--Erdős BE1
• Theorem 1.2
• Corollary 1.3
• Conjecture 1.4: Burr--Erdős BE1
• Theorem 1.6
• Corollary 1.7
• Corollary 1.8
• Corollary 1.9
• proof
• Conjecture 1.12: Burr--Erdős BE2