For any countably infinite graph G, Ramsey’s theorem guarantees an infinite monochromatic copy of G in any r-coloring of the edges of the countably infinite complete graph Kℕ. Taking this a step further, it is natural to wonder how “large” of a monochromatic copy of G we can find with respect to some measure - for instance, the density (or upper density) of the vertex set of G in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked. In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of Kℕ, there exists a monochromatic path whose vertex set has upper density at least 2/3, but it is not possible to do better than 8/9. They also showed that for some sequence en → 0, there exists a monochromatic path P such that for infinitely many n, the set (1,2,...,n contains the first $$\left(\frac{1}{3+\sqrt{3}}-\epsilon_n\right)n$$
 vertices of P, but it is not possible to do better than 2n/3. We improve both results, in the former case achieving an upper density at least 3/4 and in the latter case obtaining a tight bound of 2/3. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.

Density of Monochromatic Infinite Subgraphs

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where $$k\ge 4$$
 . It was conjectured that extremal graphs are those which have clique number k and size $${k\atopwithdelims ()2}+n-k$$
 . We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with $$k\ge 4$$
 . We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is $$K_k\cup E_{n-k}$$
 , $$K_1\vee (K_{k-1}\cup E_{n-k})$$
 and $$(K_1 \vee (K_{k-1} \cup E_{n-k-c+1}))\cup E_{c-1}$$
 respectively.

/pdf/extremal-colorings-and-independent-sets-486z71uoo9.pdf

Extremal Colorings and Independent Sets

For all positive integers $r\geq 3$ and $n$ such that $r^2-r$ divides $n$ and an affine plane of order $r$ exists, we construct an $r$-edge colored graph with minimum degree $(1-\frac{r-2}{r^2-r})n-2$ such that the largest monochromatic component has order less than $\frac{n}{r-1}$. This generalizes an example of Guggiari and Scott and, independently, Rahimi for $r=3$ and thus disproves a conjecture of Gyarfas and Sarkozy for all integers $r\geq 3$ such that an affine plane of order $r$ exists.

A note about monochromatic components in graphs of large minimum degree.

The Reconstruction Conjecture of Ulam asserts that, for $n\geq 3$, every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-1$ vertices. The conjecture is known to hold for various special classes of graphs but remains wide open. We survey results on the more general conjecture by Kelly from 1957 that for every positive integer $\ell$ there exists $M_\ell$ (with $M_1=3$) such that when $n\geq M_\ell$ every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-\ell$ vertices.

On reconstruction of graphs from the multiset of subgraphs obtained by deleting $\ell$ vertices

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.

Density of monochromatic infinite paths.

For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gy\'{a}rfas and S\'{a}rk\"{o}zy.

/pdf/monochromatic-components-in-edge-coloured-graphs-with-large-5f1dwt28f6.pdf

Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

Let $\vec{K}_{\mathbb{N}}$ be the complete symmetric digraph on the positive integers. Answering a question of DeBiasio and McKenney, we construct a 2-colouring of the edges of $\vec{K}_{\mathbb{N}}$ in which every monochromatic path has density 0. On the other hand, we show that, in every colouring that does not have a directed path with $r$ edges in the first colour, there is directed path in the second colour with density at least $\frac1r$.

Monochromatic Paths in the Complete Symmetric Infinite Digraph

The deck of a graph $G$ is given by the multiset of (unlabelled) subgraphs $\{G-v:v\in V(G)\}$. The subgraphs $G-v$ are referred to as the cards of $G$. Brown and Fenner recently showed that, for $n\geq29$, the number of edges of a graph $G$ can be computed from any deck missing 2 cards. We show that, for sufficiently large $n$, the number of edges can be computed from any deck missing at most $\frac1{20}\sqrt{n}$ cards.

Size reconstructibility of graphs

For graphs G and H, an H-colouring of G is a map ψ : V (G) → V (H) such that ij ∈ E(G) ⇒ ψ(i)ψ(j) ∈ E(H). The number of H-colourings of G is denoted by hom(G,H). We prove the following: for all graphs H and δ ≥ 3, there is a constant κ(δ,H) such that, if n ≥ κ(δ,H), the graph Kδ,n−δ maximises the number of H-colourings among all connected graphs with n vertices and minimum degree δ. This answers a question of Engbers. We also disprove a conjecture of Engbers on the graph G that maximises the number of H-colourings when the assumption of the connectivity of G is dropped. Finally, let H be a graph with maximum degree k. We show that, if H does not contain the complete looped graph on k vertices or Kk,k as a component and δ ≥ δ0(H), then the following holds: for n sufficiently large, the graph Kδ,n−δ maximises the number of Hcolourings among all graphs on n vertices with minimum degree δ. This partially answers another question of Engbers.

Hannah Guggiari

Papers

Monochromatic Components in Edge-Coloured Graphs with Large Minimum Degree

Monochromatic Paths in the Complete Symmetric Infinite Digraph

Size reconstructibility of graphs

Maximising ‐colourings of graphs

Size reconstructibility of graphs