We study colorings of Euclidean spaces avoiding specified patterns on straight lines. This extends the seminal work of Thue on avoidability properties of sequences to continuous, higher dimensional structures. We prove that every space $\mathbb{R}^d$ has a $2$-coloring such that no sequence of colors derived from collinear points separated by unit distance consists of more than $r(d)$ identical blocks. In case of the plane we show that $r(2)\leqslant 43$. We also consider more general patterns and give a sufficient condition for a pattern to be avoided in the plane. This supports a general Pattern Avoidance Conjecture in Euclidean spaces. The proofs are based mainly on the probabilistic method, but additional tools are forced by the geometric nature of the problem. We also consider similar questions for general geometric graphs in the plane. In the conclusion of the paper, we pose several conjectures alluding to some famous open problems in Euclidean Ramsey Theory.

We present results referring to the Hadwiger–Nelson problem which asks for the minimum number of colors needed to color the plane with no two points at distance 1 having the same color. Exoo considered a more general problem concerning graphs $$G_{[a,b]}$$ with $$\mathbb {R}^2$$ as the vertex set and two vertices adjacent if their distance is in the interval [a, b]. Exoo conjectured $$\chi (G_{[a,b]}) = 7$$ for sufficiently small but positive difference between a and b. We partially answer this conjecture by proving that $$\chi (G_{[a,b]}) \geqslant 5$$ for $$b > a$$ . A j-fold coloring of a graph $$G = (V,E)$$ is an assignment of j-elemental sets of colors to the vertices of G, in such a way that the sets assigned to any two adjacent vertices are disjoint. The fractional chromatic number $$\chi _f(G)$$ is the infimum of fractions k / j for j-fold coloring of G using k colors. We generalize a method by Hochberg and O’Donnel (who proved that $$G_{[1,1]} \leqslant 4.36$$ ) for the fractional coloring of graphs $$G_{[a,b]}$$ , obtaining a bound dependent on $$\frac{a}{b}$$ . We also present few specific and two general methods for j-fold coloring of $$G_{[a,b]}$$ for small j, in particular for $$G_{[1,1]}$$ and $$G_{[1,2]}$$ . The j-fold coloring for small j has strong practical motivation especially in scheduling theory, while graph $$G_{[1,2]}$$ is often used to model hidden conflicts in radio networks.

A necklace splitting theorem of Goldberg and West asserts that any $k$-colored (continuous) necklace can be fairly split using at most $k$ cuts. Motivated by the problem of Erdös on strongly nonrepetitive sequences, Alon et al. proved that there is a $(t+3)$-coloring of the real line in which no necklace has a fair splitting using at most $t$ cuts. We generalize this result for higher dimensional spaces. More specifically, we prove that there is $k$-coloring of $\mathbb{R}^{d}$ such that no cube has a fair splitting of size $t$, i.e., for each of the axes using at most $t$ hyperplanes orthogonal to it, provided $k\geq (t+4)^{d}-(t+3)^{d}+(t+2)^{d}-2^{d}+d(t+2)+3$. We also consider a discrete variant of the multidimensional necklace splitting problem in the spirit of the theorem of de Longueville and Živaljević. The question of how many axes-aligned hyperplanes are needed for a fair splitting of a $d$-dimensional $k$-colored cube remains open.