![](/files/happy5.png)
Por, Attila
Geometric Hales-Jewett Theorem ★★
Conjecture For all integers
and
, there is an integer
such that for every set
of at least
points in the plane, if each point in
is assigned one of
colours, then:
![$ k\geq1 $](/files/tex/636eac0b23e9e5ba1f1a1fc5e22e2d2009ff1533.png)
![$ \ell\geq3 $](/files/tex/06509000d5e4893e990a6bf4e2deca4af4e82a6c.png)
![$ f(k,\ell) $](/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ f(k,\ell) $](/files/tex/b9ec548be74b7eb11cfc1d7f1b688bc508002543.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
- \item
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ P $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
Keywords: Hales-Jewett Theorem; ramsey theory
Big Line or Big Clique in Planar Point Sets ★★
Let be a set of points in the plane. Two points
and
in
are visible with respect to
if the line segment between
and
contains no other point in
.
Conjecture For all integers
there is an integer
such that every set of at least
points in the plane contains at least
collinear points or
pairwise visible points.
![$ k,\ell\geq2 $](/files/tex/badf6cb8e9340567fbf08b1790403654f6fa5c2e.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Keywords: Discrete Geometry; Geometric Ramsey Theory
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