Chromatic number of Euclidean plane

If the chromatic number of Euclidean plane is larger than four, but it is known that the chromatic number of planar graphs is equal to four, then how does one explain it? In my opinion, they are contradictory to each other. This idea leads to confirm the chromatic number of the plane about its exact value

Finite ε-unit distance graphs

In 2005, Exoo posed the following question. Fix $\epsilon$ with 0\leq\epsilon<1. Let $G_\epsilon$ be the graph whose vertex set is the Euclidean plane, where two vertices are adjacent iff the Euclidean distance between them lies in the closed interval $[1-\epsilon,1+\epsilon]$. What is the chromatic number $\chi(G_\epsilon)$ of this graph? The case $\epsilon=0$ is precisely the classical ``chromatic number of the plane'' problem. In a 2018 preprint, de Grey shows that $\chi(G_0)\geq 5$; the proof relies heavily on machine computation. In 2016, Grytczuk et al. proved a weaker result with a human-comprehensible but nonconstructive proof: whenever 0<\epsilon<1, we have that $\chi(G_\epsilon)\geq 5$. (This lower bound of $5$ was later improved by Currie and Eggleton to $6$.) The De Bruijn - Erd\H{o}s theorem (which relies on the axiom of choice) then guarantees the existence, for each $\epsilon$, of a finite subgraph $H_\epsilon$ of $G_\epsilon$ such that $\chi(H_\epsilon)\geq 5$. In this paper, we explicitly construct such finite graphs $H_\epsilon$. We find that the number of vertices needed to create such a graph is no more than $2\pi(15+14\epsilon^{-1})^2$. Our proof can be done by hand without the aid of a computer

The Fractional Chromatic Number of the Plane

The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted $\chi(\mathcal{R}^2)$. The problem was introduced in 1950, and shortly thereafter it was proved that $4\le \chi(\mathcal{R}^2)\le 7$. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate $\chi_f(\mathcal{R}^2)$, the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were $3.5556 \le \chi_f(\mathcal{R}^2)\le 4.3599$. Here we improve the lower bound to $76/21\approx3.6190$.Comment: 20 pages, 10 figure