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Wood, David R.
Chromatic number of associahedron ★★
Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood
Conjecture Associahedra have unbounded chromatic number.
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 $](/files/tex/b2b0b759db4d5a1b3204c38cdee6d9bd9e0d0dab.png)
Keywords: Hales-Jewett Theorem; ramsey theory
Generalised Empty Hexagon Conjecture ★★
Author(s): Wood
Conjecture For each
there is an integer
such that every set of at least
points in the plane contains
collinear points or an empty hexagon.
![$ \ell\geq3 $](/files/tex/06509000d5e4893e990a6bf4e2deca4af4e82a6c.png)
![$ f(\ell) $](/files/tex/4d2a38aa4584bb0aecf3b85c5e0fc3f83263108c.png)
![$ f(\ell) $](/files/tex/4d2a38aa4584bb0aecf3b85c5e0fc3f83263108c.png)
![$ \ell $](/files/tex/d2c5960dd9795a1b000a5843d282c97268e303c4.png)
Keywords: empty hexagon
Colouring $d$-degenerate graphs with large girth ★★
Author(s): Wood
Question Does there exist a
-degenerate graph with chromatic number
and girth
, for all
and
?
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ d + 1 $](/files/tex/2b7e8b22bf0d4e0aa6cc7dcc6acf051bab97990f.png)
![$ g $](/files/tex/4239ee4145983e1d8ad375f0606cc7140bce36a3.png)
![$ d \geq 2 $](/files/tex/6dfe01f1d81a75b84c1284e30c319cc6137173b1.png)
![$ g \geq 3 $](/files/tex/58bd309cb5a84dc6bb041e8d02207c64d974c46d.png)
Keywords: degenerate; girth
Forcing a 2-regular minor ★★
Conjecture Every graph with average degree at least
contains every 2-regular graph on
vertices as a minor.
![$ \frac{4}{3}t-2 $](/files/tex/e3058b3a3212b3db0404690975e13ff4036eafd0.png)
![$ t $](/files/tex/4761b031c89840e8cd2cda5b53fbc90c308530f3.png)
Keywords: minors
Fractional Hadwiger ★★
Author(s): Harvey; Reed; Seymour; Wood
Conjecture For every graph
,
(a)![$ \chi_f(G)\leq\text{had}(G) $](/files/tex/50fb973c4dd31a8fde6ae9c6a9ba74c3eca2849a.png)
(b)![$ \chi(G)\leq\text{had}_f(G) $](/files/tex/6ef2849ce5271ed2eb983602db9e4948dcc30e87.png)
(c)
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
(a)
![$ \chi_f(G)\leq\text{had}(G) $](/files/tex/50fb973c4dd31a8fde6ae9c6a9ba74c3eca2849a.png)
(b)
![$ \chi(G)\leq\text{had}_f(G) $](/files/tex/6ef2849ce5271ed2eb983602db9e4948dcc30e87.png)
(c)
![$ \chi_f(G)\leq\text{had}_f(G) $](/files/tex/7e131f28dcdff73d40ea8cfaf990a2ad70fb9952.png)
Keywords: fractional coloring, minors
Forcing a $K_6$-minor ★★
Author(s): Barát ; Joret; Wood
Conjecture Every graph with minimum degree at least 7 contains a
-minor.
![$ K_6 $](/files/tex/ba39f6d33baf4500a21bc1905745d7c0414444e6.png)
Conjecture Every 7-connected graph contains a
-minor.
![$ K_6 $](/files/tex/ba39f6d33baf4500a21bc1905745d7c0414444e6.png)
Keywords: connectivity; graph minors
Point sets with no empty pentagon ★
Author(s): Wood
Problem Classify the point sets with no empty pentagon.
Keywords: combinatorial geometry; visibility graph
Number of Cliques in Minor-Closed Classes ★★
Author(s): Wood
Question Is there a constant
such that every
-vertex
-minor-free graph has at most
cliques?
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ K_t $](/files/tex/7a86d3ef1cad6ecf4b2ce1338d254d4b623a47d1.png)
![$ c^tn $](/files/tex/8ec4fa87f4377d2cbdd88cb8baab37d6198085d1.png)
Big Line or Big Clique in Planar Point Sets ★★
Let be a set of points in the plane. Two points
and
in
are visible with respect to
if the line segment between
and
contains no other point in
.
Conjecture For all integers
there is an integer
such that every set of at least
points in the plane contains at least
collinear points or
pairwise visible points.
![$ 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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