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

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 $R \in \mathcal{R}$.*

… →

[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]

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},
}
```