On a theorem of Mycielski and Taylor

$\DeclareMathOperator{\cov}{cov}\newcommand{\meager}{\mathcal{M}}$

In 1964, Jan Mycielski [1] 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 [2].

Theorem (Galvin). 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. Andreas Blass [3] 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 [4] 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.

Alan Taylor [5] generalized the result to Baire measurable colorings $c:[X]^2\to\kappa$ where $\kappa$ is any cardinal smaller than $\cov(\meager)$. 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 [6]. 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, Taylor). 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 [7].

Lemma. If $\mathcal{P}$ is a countable partial order and $\mathcal{D}$ is a family of dense subsets of $\mathcal{P}$ with $|\mathcal{D}| \lt \cov(\meager)$, 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 [7].

For simplicity, we will assume that $X$ is Baire space $\omega^\omega$. As usual, we write $$[s] = \set{x \in \omega^\omega : s \subset x}$$ 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 $|f_p(s)| \geq d_p$ 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 $$R \cap \prod_{s \in \Sigma} [f_p(s)] = \emptyset$$ 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 $$Z = \set{ x \in \omega^\omega : (\forall p \in G)(\exists s \in 2^{d_p})(f_p(s) \subset x) } = \bigcap_{p \in G} \bigcup_{s \in 2^{d_p}} [f_p(s)]$$ 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] J. Mycielski, “Independent sets in topological algebras,” Fund. math., vol. 55, pp. 139-147, 1964.
[Bibtex]
@article {Mycielski,
AUTHOR = {Mycielski, Jan},
TITLE = {Independent sets in topological algebras},
JOURNAL = {Fund. Math.},
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},
VOLUME = {55},
YEAR = {1964},
PAGES = {139--147},
ISSN = {0016-2736},
MRCLASS = {08.40 (55.99)},
MRNUMBER = {0173645 (30 \#3855)},
MRREVIEWER = {Th. J. Dekker},
URL = {http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=55},
}
[2] F. Galvin, “Partition theorems for the real line,” Notices amer. math. soc., vol. 15, p. 660, 1968.
[Bibtex]
@article {Galvin,
AUTHOR = {Galvin, Fred},
TITLE = {Partition theorems for the real line},
JOURNAL = {Notices Amer. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical Society},
VOLUME = {15},
YEAR = {1968},
PAGES = {660},
NOTE = {Erratum in volume 16 (1969), p.~1095},
}
[3] [doi] A. Blass, “A partition theorem for perfect sets,” Proc. amer. math. soc., vol. 82, iss. 2, pp. 271-277, 1981.
[Bibtex]
@article {Blass,
AUTHOR = {Blass, Andreas},
TITLE = {A partition theorem for perfect sets},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {82},
YEAR = {1981},
NUMBER = {2},
PAGES = {271--277},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {03E15 (03E05 04A20 54H05)},
MRNUMBER = {609665 (83k:03063)},
DOI = {10.2307/2043323},
URL = {http://dx.doi.org/10.2307/2043323},
}
[4] F. G. Dorais and R. Filipów, “Algebraic sums of sets in Marczewski-Burstin algebras,” Real anal. exchange, vol. 31, iss. 1, pp. 133-142, 2005/06.
[Bibtex]
@article {DoraisFilipow,
AUTHOR = {Dorais, Fran{\c{c}}ois G. and Filip{\'o}w, Rafa{\l}},
TITLE = {Algebraic sums of sets in {M}arczewski-{B}urstin algebras},
JOURNAL = {Real Anal. Exchange},
FJOURNAL = {Real Analysis Exchange},
VOLUME = {31},
YEAR = {2005/06},
NUMBER = {1},
PAGES = {133--142},
ISSN = {0147-1937},
MRCLASS = {28A05 (39A70)},
MRNUMBER = {2218194 (2007h:28002)},
MRREVIEWER = {K. P. S. Bhaskara Rao},
URL = {http://projecteuclid.org/getRecord?id=euclid.rae/1149516801},
}
[5] A. D. Taylor, “Partitions of pairs of reals,” Fund. math., vol. 99, iss. 1, pp. 51-59, 1978.
[Bibtex]
@article {Taylor,
AUTHOR = {Taylor, Alan D.},
TITLE = {Partitions of pairs of reals},
JOURNAL = {Fund. Math.},
FJOURNAL = {Polska Akademia Nauk. Fundamenta Mathematicae},
VOLUME = {99},
YEAR = {1978},
NUMBER = {1},
PAGES = {51--59},
ISSN = {0016-2736},
MRCLASS = {04A20},
MRNUMBER = {0465873 (57 \#5759)},
MRREVIEWER = {Thomas J. Jech},
URL = {http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=99},
}
[6] F. G. Dorais, R. Filipów, and T. Natkaniec, “On some properties of Hamel bases and their applications to Marczewski measurable functions,” Preprint, 2011.
[Bibtex]
@article {DoraisFilipowNatkaniec,
AUTHOR = {Dorais, Fran{\c{c}}ois G. and Filip{\'o}w, Rafa{\l} and Natkaniec, Tomasz},
TITLE = {On some properties of {H}amel bases and their applications to {M}arczewski measurable functions},
JOURNAL = {preprint},
YEAR = {2011},
}
[7] T. Bartoszyński and H. Judah, Set theory, Wellesley, MA: A K Peters Ltd., 1995.
[Bibtex]
@book {BartoszynskiJudah,
AUTHOR = {Bartoszy{\'n}ski, Tomek and Judah, Haim},
TITLE = {Set theory},
NOTE = {On the structure of the real line},
PUBLISHER = {A K Peters Ltd.},
ADDRESS = {Wellesley, MA},
YEAR = {1995},
PAGES = {xii+546},
ISBN = {1-56881-044-X},
MRCLASS = {03-02 (03Exx)},
MRNUMBER = {1350295 (96k:03002)},
MRREVIEWER = {Eva Coplakova},
}
 

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