Questions tagged [colorings]
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15 questions
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Nonrepetitive nonhomogenous partition regularity
Is it true that for every $k$ for every $k$-coloring of the natural numbers there are naturals $a_1,\dots,a_{2l}$ for some $l\ge 2$ such that $a_1+a_3=2a_2+2$, $a_2+a_4=2a_3+2$, ..., $a_{2l-2}+a_{2l}=...
- 18.7k
2
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1
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117
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A problem about the existence of increasing coloring groups
Got stuck on this one for months.
Given a sequence of non decreasing positive integers $a_1, .., a_n$, let there be $a_i$ balls labeled the number i, for each $1 \leq i \leq n$. Suppose there is a k ...
- 4,456
34
votes
1
answer
2k
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Two-colouring the two-sphere
Suppose that $S^2$ is the unit sphere in $\mathbb{R}^3$.
Is there a function $f \colon S^2 \to \{0,1\}$ so that, for any orthonormal basis $(u,v,z)$, exactly one of the values $f(u)$, $f(v)$, and $f(...
- 679
10
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310
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Coloring for arithmetic progressions with 2-power difference restricted to a set of numbers
For $D \subset \mathbb N$, let $\mathbb A_D$ be the set of all arithmetic progressions with difference in $D$ (and of finite length).
Let $\mathbb A_D$ be $m$-good if for every $S \subset \mathbb N$, ...
- 18.7k
2
votes
0
answers
121
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A variation of packing chromatic numbers for $\mathbb Z^d$
Subercaseaux and Heule showed in
https://arxiv.org/abs/2301.09757
(The Packing Chromatic Number of the Infinite Square Grid is 15)
that $n=15$ is the smallest positive integer for which there
is a map ...
- 17.9k
14
votes
5
answers
2k
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How can one construct a four-coloring of a tiling of the plane with Smith, Myers, Kaplan, and Goodman-Strauss's aperiodic monotile?
This is motivated by the new paper of Smith, Myers, Kaplan, and Goodman-Strauss, wherein they define the existence of an aperiodic monotile. Clearly their tiling is not three-colorable, so we have ...
- 181
2
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1
answer
166
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$n^2$-Grid $3n$-Coloring Game: Can we color a n-square grid with 3n colors s. t. we can't select n colors to get an histogram with $\Theta(n^2)$ area?
The coloring game is a game played between Alice and Bob.
There exists a grid of size $n \times n$, where $n$ is a strictly positive integer.
Each cell of the grid can be colored with a color that ...
- 21
4
votes
2
answers
192
views
Where can I find a picture of the complete 9-map on a triple torus that corresponds to Heffter’s table?
What I’m looking for is the analogue of
Figure 5 in the paper by Saul Stahl, The Othe Map Coloring Theorem, Mathematics Magazine 1985, which is a complete 8-map $M_8$ on the double torus $S_2$ that ...
- 111
6
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0
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333
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What do the circuits of this matroid look like?
Given any hypergraph $H=(V,E)$ we call a family of sets $I\subseteq E$ "indifferent" iff there exists a map $\phi:I\to V$ such that: $\forall X\in I(\phi(X)\in X)$ and $\forall X,Y\in I(X\...
- 2,506
1
vote
1
answer
216
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Maximum number of colors for an optimal tiling which guarantees infinite paths
This question is a more applicable version of the question I've asked in mathexchange recently:
What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square
block ...
- 289
2
votes
0
answers
86
views
How many ways to cover a N×N chessboard with white and black boxes by some restrictions?
Suppose we have a N×N chessboard and the boxes ■, □.
We should cover the chessboard with those boxes but there can not have the 2×2 square $\scriptstyle{\begin{array}{cc}\square&\square\\
\...
- 33
3
votes
1
answer
113
views
What number of colorings can guarantee that for every k-element subset there exists a coloring assigns different colors for elements from this subset?
Let $M(n, k)$ be a minimal number $m$ such that there exists set $C$ ($|C|=m$) of colorings of n-element set $[n]$ with $k$ colors such that for every $k$-element subset $K$ of $[n]$ there exists ...
- 33
4
votes
2
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265
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Can different knots have the same numbers of quandle colorings for all quandles?
Let $K_1$ and $K_2$ be two knots such that for all finite quandles $X$, the number of colorings of $K_1$ by $X$ is the same as the number of colorings of $K_2$ by $X$. Then my question is, must $K_1$ ...
- 4,587
4
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0
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63
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Bipartite graphs with alternating edge colorings
I would like to find all graphs or lattices which satisfy the following conditions:
(1) Graph is bipartite with vertex types $A$ and $B$ ($A$-vertices only connected to $B$-vertices and vice-versa)
(...
2
votes
1
answer
297
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Homogeneous van der Waerden
The Erdős Discrepancy Problem is whether in any two-coloring of the naturals for any $C$ there is a sequence $d, 2d, \ldots nd$ such that the difference of red and blue numbers in it is more than $C$.
...
- 18.7k