If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ordertype $\alpha$. Generalizing this to $[\lambda]^{<\kappa}$, we say that a subset $S\subseteq[\lambda]^{<\kappa}$ (recall that this is the space of all subsets of $\lambda$ of size ${<}\,\kappa$) is fat if for every $\Theta\geq\lambda$, every $\alpha<\kappa$ and every club $C\subseteq[H(\Theta)]^{<\kappa}$ there is a continuous sequence $(N_i)_{i<\alpha}$ of elements of $C$ such that for every $i$, $N_i\in N_{i+1}$ and $N_i\cap\lambda\in S$. Just like for “regular” fatness, a set $S$ being fat is equivalent to saying that $S$ can obtain a club subset in an extension by a ${<}\,\kappa$-distributive forcing.
However, for $[\lambda]^{<\kappa}$ (if $\lambda>\kappa$), we can get away with a stronger property: Let us call a set $S\subseteq[\lambda]^{<\kappa}$ very fat if for every $\Theta\geq\lambda$ and every club $C\subseteq[H(\Theta)]^{<\kappa}$ there is a continuous sequence $(N_i)_{i<\kappa}$ of elements of $C$ such that for every $i$, $N_i\in N_{i+1}$ and $N_i\cap\lambda\in S$. This is equivalent to saying that $S$ can obtain a club subset in an extension by a forcing $\mathbb{P}$ such that for every sequence $(D_i)_{i<\kappa}$ of open dense sets in $\mathbb{P}$, there is a descending sequence $(p_i)_{i<\kappa}$ such that $p_i\in D_i$ for every $i<\kappa$ (which is an interesting property in and of itself).
It turns out that it is consistent that there are very fat subsets of $[\lambda]^{<\kappa}$ not containing a club if (and only if) $\lambda>\kappa$: By a result of Gitik, after forcing with (e.g.) $\operatorname{Add}(\omega)$, for any regular uncountable $\kappa<\lambda$, the set $[\lambda]^{<\kappa}\smallsetminus V$ is stationary, so $[\lambda]^{<\kappa}\cap V$ does not contain a club. However, by the ccc. of $\operatorname{Add}(\omega)$, the set remains very fat. This leaves open the following questions:
- Is it consistent that for every $\kappa<\lambda$, every very fat subset of $[\lambda]^{<\kappa}$ contains a club (the critical case seems to be $\kappa=\omega_1$)? In particular, what happens in $L$?
- Is it consistent that for some $\kappa$ and $\lambda$, there are two disjoint very fat subsets of $[\lambda]^{<\kappa}$? Again, what is the situation in $L$?
