(Pierre Gillibert asked me this question and I post it with his permission.)

Let $X$ be an infinite set, and $f\colon[X]^\omega\to[X]^\omega$. We say that $\{x_n\mid n<\omega\}$ is a free decreasing sequence (for $f$) if for all $n$, $x_n\notin f(\{x_k\mid k>n\})$.

Is there an infinite set $X$ such that for every isotone $f\colon[X]^\omega\to[X]^\omega$ there exists a free decreasing sequence? Is it at least consistent from assumptions such as $V=L$, large cardinals or strong forcing axioms?

Some observations:

1. It is clear that $X$ is uncountable, because otherwise $f(A)=X$ for all $A\in[X]^\omega$ would pose a counterexample.

2. If there is no such set of size $\kappa$, then there is no such example of size $\kappa^+$.

3. If $\kappa$ has uncountable cofinality, and there is no such set of size $<\kappa$, then there is no example of size $\kappa$, since we can "glue" counterexamples and use the fact that every countable set is bounded (this is in effect the same proof for the previous observation).

(Pierre Gillibert asked me this question and I post it with his permission.)

Let $X$ be an infinite set, and $f\colon[X]^\omega\to[X]^\omega$. We say that $\{x_n\mid n<\omega\}\subseteq X$ is a free decreasing sequence (for $f$) if for all $n$, $x_n\notin f(\{x_k\mid k>n\})$.

Is there an infinite set $X$ such that for every isotone $f\colon[X]^\omega\to[X]^\omega$ there exists a free decreasing sequence? Is it at least consistent from assumptions such as $V=L$, large cardinals or strong forcing axioms?

Some observations:

1. It is clear that $X$ is uncountable, because otherwise $f(A)=X$ for all $A\in[X]^\omega$ would pose a counterexample.

2. If there is no such set of size $\kappa$, then there is no such example of size $\kappa^+$.

3. If $\kappa$ has uncountable cofinality, and there is no such set of size $<\kappa$, then there is no example of size $\kappa$, since we can "glue" counterexamples and use the fact that every countable set is bounded (this is in effect the same proof for the previous observation).