Coloring some finite sets in {R}^{n}

Coloring Some Finite Sets In

This note relates to bounds on the chromatic number χ(R) of the Euclidean space, which is the minimum number of colors needed to color all the points in R so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in R n was introduced showing that χ(R) ≥ χ(Gn) ≥ (1 + o(1)) n 2 6 . For many years, this bound has been remaining the best known bound for the ...

Coloring Some Finite Sets in R

This note relates to bounds on the chromatic number χ(R) of the Euclidean space, which is the minimum number of colors needed to color all the points in R so that any two points at the distance 1 receive different colors. In [6] a sequence of graphs Gn in R was introduced showing that χ(R) ≥ χ(Gn) ≥ (1 + o(1)) 2 6 . For many years, this bound has been remaining the best known bound for the chro...

Coloring finite subsets of uncountable sets

It is consistent for every 1 ≤ n < ω that 2 = ωn and there is a function F : [ωn] <ω → ω such that every finite set can be written at most 2 − 1 ways as the union of two distinct monocolored sets. If GCH holds, for every such coloring there is a finite set that can be written at least 1 2 ∑n i=1 ( n+i n )( n i ) ways as the union of two sets with the same color. 0 Introduction In [6] we proved ...

A sequential coloring algorithm for finite sets

Let X be a finite set and P a hereditary property associated with the subsets of X. A partition of X into n subsets each with property P is said to be a P-n-coloring of X. The minimum n such that a P-n-coloring of X exists is defined to be the P-chromatic number of X. In this paper we give a sequential coloring algorithm for P-coloring X. From the algorithm we then get a few upper bounds for th...

CONVEXITY NUMBERS OF CLOSED SETS IN Rn