Por, Attila


Geometric Hales-Jewett Theorem ★★

Author(s): Por; Wood

\begin{conjecture} For all integers $k\geq1$ and $\ell\geq3$, there is an integer $f(k,\ell)$ such that for every set $P$ of at least $f(k,\ell)$ points in the plane, if each point in $P$ is assigned one of $k$ colours, then: \begin{itemize} \item $P$ contains $\ell$ collinear points, or \item $P$ contains a monochromatic line (that is, a maximal set of collinear points receiving the same colour) \end{itemize} \end{conjecture}

Keywords: Hales-Jewett Theorem; ramsey theory

Big Line or Big Clique in Planar Point Sets ★★

Author(s): Kara; Por; Wood

Let $S$ be a set of points in the plane. Two points $v$ and $w$ in $S$ are \emph{visible} with respect to $S$ if the line segment between $v$ and $w$ contains no other point in $S$.

\begin{conjecture} For all integers $k,\ell\geq2$ there is an integer $n$ such that every set of at least $n$ points in the plane contains at least $\ell$ collinear points or $k$ pairwise visible points. \end{conjecture}

Keywords: Discrete Geometry; Geometric Ramsey Theory

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