Let denote the collection of infinite subsets of (the first infinite ordinal) and let be any map. We say that is monochromatic with respect to if restricted to is constant (or, equivalently, if for all with ). We say is Ramsey if there is such that is monochromatic with respect to .
Certainly, plenty of examples of Ramsey functions come to mind, which leads to the question:
Are there non-Ramsey functions ?
It is indeed a bit tricky to come up with one – the construction I present next needs the Axiom of Choice ( https://en.wikipedia.org/wiki/Axiom_of_choice ). We define a binary relation on by
if and only if the symmetric difference is finite.
From “the infinitist’s point of view” two sets are the same if they only differ in finitely many elements. It is easy to check that this an equivalence relation.
Let be the set of equivalence classes of with respect to the equivalence relation . We use the Axiom of Choice to find a set of representatives such that
for all blocks .
We define the representative function by assigning to each the unique element such that .
Finally, we define by setting for if is even, and otherwise. It is easy to see that whenever , then is not monochromatic: As is finite and and are both infinite, pick . So the parity of
and
is not the same, therefore restricted to is not constant.
Reformulation with property
A hypergraph is a pair where is a non-empty set and . We say that has property if there is such that for all . (In other words, intersects every , but no is fully contained in .)
The question whether there is a non-Ramsey function is equivalent to asking whether the hypergraph where
has property .
Further questions
- Does the statement “ has property ” imply the Axiom of Choice? Or does it imply a weaker statement than AC? Or is it provable in ZF alone?
- Consider the hypergraph where is built in an analogous manner to the edge set in hypergraph . Does have property , no matter whether we assume the Axiom of Choice?
- What is in an example of a poset with no minimal elements such that the hypergraph does not have property , where is the collection of principal down-sets of ? (In other words, has the property that whenever its elements are colored red or blue, then there is a principal down-set consisting of more than 1 element such that all the elements have the same color.)