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Monroe Eskew
Monroe Eskew
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Let $\kappa$ be an infinite cardinal. Consider the following example 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 a counterexamplean 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 $n>3$?

Let $\kappa$ be an infinite cardinal. Consider the following example 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 a counterexample 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 $n>3$?

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 $n>3$?

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Ioannis Souldatos
Ioannis Souldatos
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Example to $2^\kappa\nrightarrow (3)^2_\kappa$, plus closed walks of odd length?

Let $\kappa$ be an infinite cardinal. Consider the following example 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 a counterexample 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 $n>3$?