On a theorem of Mycielski and Taylor
Mycielski (1964) proved a wonderful theorem about independent sets in Polish spaces. He showed that if is an uncountable Polish space and is a meager subset of for each , then there is a perfect set such that whenever are distinct elements of . In other words, is -independent for each . This is a wonderfully general theorem that has a multitude of applications.
One of my favorite theorems where Mycielski’s Theorem comes in handy is a remarkable partition theorem due to Fred Galvin.
Theorem (Galvin 1968). Let be an uncountable Polish space and let be a Baire measurable coloring where is a positive integer. Then has a perfect -homogeneous subset.
Galvin proved a similar result for colorings of , but with a weaker conclusion that triples from the perfect set take on at most two colors. Blass (1981) then extended Galvin’s result to colorings of , showing that there is a perfect set that takes on at most colors.
Galvin’s result has many applications too. For example, Rafał Filipów and I used it in (Dorais–Filipów 2005) it to show that if is a perfect Abelian Polish group, then contains a Marczewski null set such that the algebraic sum is not Marczewski measurable.
Taylor (1978) generalized the result to Baire measurable colorings where is any cardinal smaller than . Doing so, Taylor similarly generalized Mycielski’s Theorem, but he only stated the result for binary relations. Recently, Rafał Filipów, Tomasz Natkaniec and I needed this generalization for relations of arbitrary arity. Unfortunately, the Mycielski–Taylor result has never been stated in full generality, so we included a proof in our paper (Dorais–Filipów–Natkaniec 2013). I am copying this proof here because I think the result is of independent interest and our proof is a nice application of Cohen forcing. A nice consequence of this extended Mycielski–Taylor Theorem is that Blass’s result extends to Baire measurable colorings where in the same way that Taylor generalized Galvin’s result for partitions of pairs.
Theorem (Mycielski 1964; Taylor 1978). Let be an uncountable Polish space and let be a family of fewer than closed nowhere dense relations on , i.e., each is a closed nowhere dense subset of for some . Then contains a perfect set which is -independent for every .
Our proof relies on the following forcing characterization of , which can be found in (Bartoszyński–Judah 1995).
Lemma. If is a countable partial order and is a family of dense subsets of with , then there is a filter on that meets every element of .
In other words, , in the notation of (Bartoszyński–Judah 1995).
For simplicity, we will assume that is Baire space . As usual, we write
We may assume that the family at least contains the diagonal . We may also assume that all relations are symmetric. (Otherwise, replace each by the relation , where and denotes the set of all permutations of .)
Consider the partial order whose conditions are pairs where and is such that for all ; the ordering of is defined by iff and for all .
For and , consider the set of all conditions such that and
for all -element subset of . (Note that this condition is slightly ambiguous since no ordering of is given, but since is assumed to be symmetric any ordering will do.)
We claim is always dense in . To see this fix a condition . We may assume that . Fix an enumeration of all -element subsets of and successively define conditions in such a way that and for every . This is always possible since is closed nowhere dense. Then, is the desired extension of in .
By the Lemma, there is a filter over that meets all dense sets for and . We claim that the set
is as required.
Note that when is the diagonal relation, then if and only if and the clopen sets are pairwise disjoint for . It follows that is a perfect set.
Now, we show that is -independent for each . Let be distinct, where is the arity of . There is such that for distinct . Let . For every there are with . Since and , are pairwise distinct. Let . Since , In particular, .
T. Bartoszyński, H. Judah, 1995: Set Theory: On the structure of the real line, A K Peters, Ltd. (Wellesley, MA).
A. Blass, 1981: A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82, no. 2, 271–277.
F. G. Dorais, R. Filipów, 2005: Algebraic sums of sets in Marczewski-Burstin algebras, Real Anal. Exchange 31, no. 1, 133–142.
F. G. Dorais, R. Filipów, T. Natkaniec, 2013: On some properties of Hamel bases and their applications to Marczewski measurable functions, Central European Journal of Mathematics 11, no. 3, 487–508.
F. Galvin, 1968: Partition theorems for the real line, Notices Amer. Math. Soc. 15, 660.
J. Mycielski, 1964: Independent sets in topological algebras, Fund. Math. 55, no. 2, 139–147.
A. Taylor, 1978: Partitions of pairs of reals, Fund. Math. 99, no. 1, 51–59.