Nonrepetitive Colourings of Planar Graphs with O(log⁡n)O(\log n) Colours

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the minimum integer $k$ such that $G$ has a nonrepetitive $k$-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is $O(\sqrt{n})$ for $n$-vertex planar graphs. We prove a $O(\log n)$ upper bound

Collinear subsets of lattice point sequences—An analog of Szemerédi's theorem

AbstractSzemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ⩾ n(k, B) and 0 < a1 < … < an is a sequence of integers with an ⩽ Bn, then some k of the ai form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ⩾ m(k, B) and u0, u1, …, um is a sequence of plane lattice points with ∑i=1m…ui − ui−1… ⩽ Bm, then some k of the ui are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result

Large subsets of discrete hypersurfaces in Zd\mathbb{Z}^d contain arbitrarily many collinear points

In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence from having bounded gaps to having gaps bounded on average. We give a higher dimensional generalization of these results. Our main theorem is the following. Theorem: Let $d\in\mathbb{N}$, let $f:\mathbb{Z}^d\to\mathbb{Z}^{d+1}$ be a Lipschitz map and let $A\subset\mathbb{Z}^d$ have positive upper Banach density. Then $f(A)$ contains arbitrarily many collinear points. Note that Pomerance's theorem corresponds to the special case $d=1$. In our proof, we transfer the problem from a discrete to a continuous setting, allowing us to take advantage of analytic and measure theoretic tools such as Rademacher's theorem.Comment: 16 pages, small part of the argument clarified in light of suggestions from the refere