  Given a probability space $(X, {\cal B}, m)$, measure preserving
transformations $g_1, \dots , g_k$ of $X$, and a colour set $C$, a colouring
rule is a way to colour the space with $C$ such that the colours allowed for
apoint $x$ are determined by that point's location and the colours of the
finitely $g_1 (x), \dots , g_k(x)$ with $g_i(x) \not= x$ for all $i$ and almost
all $x$. We represent a colouring rule as a correspondence $F$ defined on
$X\times C^k$ with values in $C$. A function $f: X\rightarrow C$ satisfies the
rule at $x$ if $f(x) \in F( x, f(g_1 x), \dots , f(g_k x))$. A colouring rule
is paradoxical if it can be satisfied in some way almost everywhere with
respect to $m$, but not in {\bf any} way that is measurable with respect to a
finitely additive measure that extends the probability measure $m$ defined on
${\cal B}$ and for which the finitely many transformations $g_1, \dots , g_k$
remain measure preserving. Can a colouring rule be paradoxical if both $X$ and
the colour set $C$ are convex and compact sets and the colouring rule says if
$c: X\rightarrow C$ is the colouring function then the colour $c(x)$ must lie
($m$ a.e.) in $F(x, c(g_1(x) ), \dots , c(g_k(x)))$ for a non-empty
upper-semi-continuous convex-valued correspondence $F$ defined on $X\times
C^k$? The answer is yes, and we present such an example. We show that this
result is robust, including that any colouring that approximates the
correspondence by $\epsilon$ for small enough positive $\epsilon$ also cannot
be measurable in the same finitely additive way. Because non-empty
upper-semi-continuous convex-valued correspondences on Euclidean space can be
approximated by continuous functions, there are paradoxical colouring rules
that are defined by continuous functions.

Combinatorics