set theory - Pigeonhole Principle for infinite case

First, let me improve upon the countable counterexample of
Darij Grinberg by giving an uncountable counterexample Y.
Indeed, I shall give finite sets X_n and a subset Y of the product ΠX_n having size continuum (that is, as large as possible), such that any two distinct y, y' in Y
have only finitely many common values.

Let X_n have 2ⁿ elements, consisting
of the binary sequences of length n. Now, for each infinite
binary sequence s, let y_s be sequence in the
product ΠX_n whose n^th value is
s|n, the length n initial segment of s. Let Y consist of
all these y_s. Since there are continuum many s,
it follows that Y has size continuum.

Note that if s and t are distinct binary sequences, then eventually
the initial segments of s and t disagree. Thus, eventually,
the values of y_s and y_t are
different. Thus, y_s and y_t have only
finitely many common values. So Y is very large counterexample, as
desired.

A similar argument works still if the X_n grow
more slowly in size, as long as liminf|X_n| =
infinity. One simply spreads the construction out a bit
further, until the size of the X_i is large
enough to accommodate the same idea. That is, if the liminf
of the sizes of the X_n's is infinite, then one
can again make a counterexample set Y of size continuum.

In contrast, in the remaining case, there are no infinite
counterexamples. I claim that if infinitely many
X_n have size at most k and Y is a
subset of ΠX_n having k+1 many elements, then
there are distinct y,y' in Y having infinitely many common
values. To see this, suppose that Y has the property that
distinct y, y' in Y have only finitely many common values.
In this case, any two y, y' must eventually have different
values. So if Y has k+1 many elements, then eventually for
sufficiently large n, these k+1 many sequences in Y must be
taking on different values in every X_n. But
since unboundedly often there are only k possible values in
X_n, this is impossible.

In summary, the situation is as follows:

Theorem. Suppose that X_n is finite and
nonempty.

• If liminf |X_n| is infinite, then there is Y
  subset ΠX_n of size continuum, such that
  distinct y, y' in Y have only finitely many values in
  common.


• Otherwise, infinitely many X_n have size at most k for some k, and in this case, every Y subset
  ΠX_n of size k+1 has distinct y,y' in Y with infinitely
  many common values.

In particular, if the X_n become increasingly
large in size, then there are very bad counterexamples to the
question, and if the X_n are infinitely often
bounded in size, then there is a very strong positive answer to
the question.