There is no short and easy answer which is uncontroversial. Metaphysical presumptions lead to different answers with different levels of sophistication of theory based on such perspectives as compatibilists and non-compatibilists and variants thereof. The nature of causality and events is also controversial.
Take a step back and parse what's really happening here. Ramsey's Theorem is a combinatorial theorem which essentially says that there exists a class of mathematical objects under the right conditions such that the freedom regarding the variables of edge-labeling is independent of the existence of a particular type of a monochromatic subgraph. That means, if given all permutations possible of the sufficiently large complete graph, no matter which permutation one pulls out of the metaphorical bag of choices, one is bound to get a graph with a monochromatic clique (in the general case).
If one uses a typical definition of free will, one is presented with:
Free will is the ability to choose between different possible courses of action unimpeded.
What you are asking has a few things that require unpacking first.
The theorem in question is a product of deduction. A deduction by definition says that if the hypothesis is true, than the conclusion is true, and in this case, the theorem says more specifically that any hypothesis chosen (a distinct permutation), the conclusion is the same (a monochromatic clique). But the question springs to mind, if every choice leads to the same consequence, is it really a choice? Henry Ford is quoted as saying that "A customer can have a car painted any color he wants as long as it’s black”. This is known eponymously as Hobson's choice. But what does it mean to make a choice? One interesting finding in cognitive science is that the brain reflects a choice before an individual's phenomenological awareness thereof.
What can be taken from this intuitionally is that there is and is not an element of free will involved depending on context of theory. A person can choose not to choose, and that is a choice; and in your example, if a person chooses a permutation, she will wind up with a unique graph by definition; but that that graph also contains some commonality with all graphs means that the person is not free to choose a graph from the set without a monochromatic clique. So, whether or not one views this as an act of free will is contingent upon how the situation is framed AND one's metaphysical presumptions about the definitions and relationships of and between free will and determinism.
Part of the consideration you need give to the question is that the essence of determinism is up for grabs too! The immaterialist George Berkeley and eliminative materialist Daniel Dennett all have very different views on what constitutes determinism to begin with.
One simple example of how complicated things might be that if you accept Cartesian dualism, there is no causality between the material and mental. Can one say that the mental choice (if it really is a choice) to roll a dice (a physical object) leads to an outcome which requires apprehension of the abstract? Is the dice roll predetermined, and how would you know? A theologian would argue divine revelation makes it part of God's predetermination. Would someone who rejects supernaturalism accept that?
This issue has been without consensus since time barely memorial; it will likely stay active and vigorously debated.