Let $\kappa$ be an infinite cardinal. Consider the following example to $2^\kappa\nrightarrow (3)^2_\kappa$.

$V$ is a set of vertices, each of which is an element of $2^\kappa$. Color the edge between two vertices $f,g\in 2^\kappa$ by the least ordinal on which $f,g$ disagree. It follows that there is no homogeneous set of size $3$, i.e. there is no triangle with all three edges the same color. 

More is true: There is no closed walk of length 3,5,7,... which is monochromatic, i.e. all edges are the same color.

**Definition**:A closed walk consists of a sequence of vertices starting and ending at the same vertex.

**Question**: Can you find an example to $2^\kappa\nrightarrow (3)^2_\kappa$, therefore no monochromatic triangles are allowed, but which has closed walks of length $n$, for any odd number $n>3$?