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Impartial coloring games
Coloring games are combinatorial games where the players alternate painting
uncolored vertices of a graph one of colors. Each different ruleset
specifies that game's coloring constraints. This paper investigates six
impartial rulesets (five new), derived from previously-studied graph coloring
schemes, including proper map coloring, oriented coloring, 2-distance coloring,
weak coloring, and sequential coloring. For each, we study the outcome classes
for special cases and general computational complexity. In some cases we pay
special attention to the Grundy function
On the dimension growth of groups
Dimension growth functions of groups have been introduced by Gromov in 1999.
We prove that every solvable finitely generated subgroups of the R. Thompson
group has polynomial dimension growth while the group itself, and some
solvable groups of class 3 have exponential dimension growth with exponential
control. We describe connections between dimension growth, expansion properties
of finite graphs and the Ramsey theory.Comment: 20 pages; v3: Erratum and addendum included as Section 9. We can only
prove that the lower bound of the dimension growth of is exp sqrt(n). New
open questions and comments are added. v4: The paper is completely revised.
Dimension growth with control is introduced, connections with graph expansion
and Ramsey theory are include
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