Abstract
A factorisation
x
=
u
1
u
2
⋯
$x = u_1 u_2 \cdots$
of an infinite word
x
$x$
on alphabet
X
$X$
is called ‘monochromatic’, for a given colouring of the finite words
X
∗
$X^*$
on alphabet
X
$X$
, if each
u
i
$u_i$
is the same colour. Wojcik and Zamboni proved that the word
x
$x$
is periodic if and only if for every finite colouring of
X
∗
$X^*$
there is a monochromatic factorisation of
x
$x$
. On the other hand, it follows from Ramsey's theorem that, for any word
x
$x$
, for every finite colouring of
X
∗
$X^*$
there is a suffix of
x
$x$
having a monochromatic factorisation. A factorisation
x
=
u
1
u
2
⋯
$x = u_1 u_2 \cdots$
is called ‘super‐monochromatic’ if each word
u
k
1
u
k
2
⋯
u
k
n
$u_{k_1} u_{k_2} \cdots u_{k_n}$
, where
k
1
<
⋯
<
k
n
$k_1 < \cdots < k_n$
, is the same colour. Our aim in this paper is to show that a word
x
$x$
is eventually periodic if and only if for every finite colouring of
X
∗
$X^*$
there is a suffix of
x
$x$
having a super‐monochromatic factorisation. Our main tool is a Ramsey result about alternating sums that may be of independent interest.