# François G. Dorais

## On a theorem of Mycielski and Taylor

Mycielski (1964) proved a wonderful theorem about independent sets in Polish spaces. He showed that if $X$ is an uncountable Polish space and $R_n$ is a meager subset of $X^n$ for each $n \geq 1$, then there is a perfect set $Z \subseteq X$ such that $(z_1,\dots,z_n) \notin R_n$ whenever $z_1,\dots,z_n$ are distinct elements of $Z$. In other words, $Z$ is $R_n$-independent for each $n \geq 1$. 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 $X$ be an uncountable Polish space and let $c:[X]^2\to\set{0,\dots,k-1}$ be a Baire measurable coloring where $k$ is a positive integer. Then $X$ has a perfect $c$-homogeneous subset.

Galvin proved a similar result for colorings of $[X]^3$, 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 $[X]^n$, showing that there is a perfect set that takes on at most $(n-1)!$ 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 $X$ is a perfect Abelian Polish group, then $X$ contains a Marczewski null set $A$ such that the algebraic sum $A + A$ is not Marczewski measurable.

Taylor (1978) generalized the result to Baire measurable colorings $c:[X]^2\to\kappa$ where $\kappa$ is any cardinal smaller than $\DeclareMathOperator{\cov}{cov}\newcommand{\meager}{\mathcal{M}}\cov(\meager)$. 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 $c:[X]^n\to\kappa$ where $\kappa \lt \cov(\meager)$ in the same way that Taylor generalized Galvin’s result for partitions of pairs.

Theorem (Mycielski 1964; Taylor 1978). Let $X$ be an uncountable Polish space and let $\mathcal{R}$ be a family of fewer than $\cov(\meager)$ closed nowhere dense relations on $X$, i.e., each $R \in \mathcal{R}$ is a closed nowhere dense subset of $X^n$ for some $n = n(R)$. Then $X$ contains a perfect set which is $R$-independent for every $R \in \mathcal{R}$.

Our proof relies on the following forcing characterization of $\cov(\meager)$, which can be found in (Bartoszyński–Judah 1995).

Lemma. If $\mathcal{P}$ is a countable partial order and $\mathcal{D}$ is a family of dense subsets of $\mathcal{P}$ with $\vert \mathcal{D} \vert \lt \cov(\meager)$, then there is a filter on $\mathcal{P}$ that meets every element of $\mathcal{D}$.

In other words, $\cov(\meager) = \mathfrak{m}(\text{Cohen})$, in the notation of (Bartoszyński–Judah 1995).

For simplicity, we will assume that $X$ is Baire space $\omega^\omega$. As usual, we write

for $s \in \omega^{\lt\omega}$.

We may assume that the family $\mathcal{R}$ at least contains the diagonal $\set{(x,x): x \in \omega^\omega}$. We may also assume that all relations $R\in\mathcal{R}$ are symmetric. (Otherwise, replace each $R \in \mathcal{R}$ by the relation $\bigcup_{\sigma\in\mathrm{Sym}(n)} \set{ (x_{\sigma(1)},\dots, x_{\sigma(n)}): (x_1,\dots, x_n)\in R}$, where $n=n(R)$ and $\mathrm{Sym}(n)$ denotes the set of all permutations of $\set{1,2,\dots,n}$.)

Consider the partial order $\mathcal{P}$ whose conditions are pairs $p = (d_p,f_p)$ where $d_p \in \omega$ and $f_p:2^{d_p}\to\omega^{\lt\omega}$ is such that $\vert f_p(s) \vert \geq d_p$ for all $s \in 2^{d_p}$; the ordering of $\mathcal{P}$ is defined by $p \leq q$ iff $d_p \leq d_q$ and $f_p(s{\upharpoonright}d_p) \subseteq f_q(s)$ for all $s \in 2^{d_q}$.

For $k \in \omega$ and $R \in \mathcal{R}$, consider the set $\mathcal{D}_{k,R}$ of all conditions $p \in \mathcal{P}$ such that $k \leq d_p$ and

for all $n(R)$-element subset $\Sigma$ of $2^{d_p}$. (Note that this condition is slightly ambiguous since no ordering of $\Sigma$ is given, but since $R$ is assumed to be symmetric any ordering will do.)

We claim $\mathcal{D}_{k,R}$ is always dense in $\mathcal{P}$. To see this fix a condition $p \in \mathcal{P}$. We may assume that $d_p \geq k$. Fix an enumeration $\Sigma_1,\dots,\Sigma_m$ of all $n(R)$-element subsets of $2^{d_p}$ and successively define conditions $p \leq p_1 \leq \cdots \leq p_m$ in such a way that $d_p = d_{p_1} = \cdots = d_{p_m}$ and $R \cap \prod_{s\in\Sigma_i} [f_{p_i}(s)] = \emptyset$ for every $i = 1,\dots,m$. This is always possible since $R$ is closed nowhere dense. Then, $p_m$ is the desired extension of $p$ in $\mathcal{D}_{k,R}$.

By the Lemma, there is a filter $G$ over $\mathcal{P}$ that meets all dense sets $\mathcal{D}_{k,R}$ for $k \in \omega$ and $R \in \mathcal{R}$. We claim that the set

is as required.

Note that when $R$ is the diagonal relation, then $p \in D_{k,R}$ if and only if $d_p \geq k$ and the clopen sets $[f_p(s)]$ are pairwise disjoint for $s \in 2^{d_p}$. It follows that $Z$ is a perfect set.

Now, we show that $Z$ is $R$-independent for each $R\in\mathcal{R}$. Let $z_1,\dots,z_n \in Z$ be distinct, where $n = n(R)$ is the arity of $R$. There is $k\in\omega$ such that $z_i\restriction k \neq z_j\restriction k$ for distinct $i,j=1,\dots,n$. Let $p\in G \cap D_{k,R}$. For every $i=1,\dots,n$ there are $s_i\in 2^{d_p}$ with $z_i\in[f_p(s_i)]$. Since $d_p\geq k$ and $z_i\restriction d_p \subset f_p(s_i)$, $s_1,\dots,s_n$ are pairwise distinct. Let $\Sigma=\set{s_1,\dots,s_n}$. Since $p\in D_{k,R}$, $R \cap \prod_{s \in \Sigma} [f_p(s)] = \emptyset.$ In particular, $(z_1,\dots,z_n) \notin R$.

#### References

1. T. Bartoszyński, H. Judah, 1995: Set Theory: On the structure of the real line, A K Peters, Ltd. (Wellesley, MA).

2. A. Blass, 1981: A partition theorem for perfect sets, Proc. Amer. Math. Soc. 82, no. 2, 271–277.

3. F. G. Dorais, R. Filipów, 2005: Algebraic sums of sets in Marczewski-Burstin algebras, Real Anal. Exchange 31, no. 1, 133–142.

4. 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.

5. F. Galvin, 1968: Partition theorems for the real line, Notices Amer. Math. Soc. 15, 660.

6. J. Mycielski, 1964: Independent sets in topological algebras, Fund. Math. 55, no. 2, 139–147.

7. A. Taylor, 1978: Partitions of pairs of reals, Fund. Math. 99, no. 1, 51–59.

Originally posted on by François G. Dorais. To the extent possible under law, François G. Dorais has waived all copyright and neigboring rights to this work.