  In 2006 T. Brown asked the following question: Given a non-periodic infinite
word $x=x_1x_2x_3\cdots$ with values in a non-empty set $\mathbb{A},$ does
there exist a finite coloring $\varphi: \mathbb{A}^+\rightarrow C$ relative to
which $x$ does not admit a $\varphi$-monochromatic factorisation, i.e., a
factorisation of the form $x=u_1u_2u_3\cdots$ with $\varphi(u_i)=\varphi(u_j)$
for all $i,j\geq 1$? Various partial results in support of an affirmative
answer to this question have appeared in the literature in recent years. In
particular it is known that the question admits an affirmative answer for all
non-uniformly recurrent words and various classes of uniformly recurrent words
including Sturmian words. In this note we answer this question in general by
showing that if $x=x_1x_2x_3\cdots$ is an infinite word with values in a
non-empty set $\mathbb{A},$ then $x$ is periodic if and only if for every
$2$-coloring $\varphi: \mathbb{A}^+\rightarrow \{0,1\}$ there exists a
$\varphi$-monochromatic factorisation of $x.$ This characterization of
periodicity of infinite words may be reformulated in the language of
ultrafilters. Let $\beta \mathbb{A}^+$ denote the Stone-Cech compactification
of the discrete semigroup $\mathbb{A}^+$ which we regard as the set of all
ultrafilters on $\mathbb{A}^+.$ Then $x$ is periodic if and only if there
exists $p\in \beta \mathbb{A}^+$ such that for each $A\in p$ there exists a
factorisation $x=u_1u_2u_3\cdots $ with each $u_i \in A.$
