$\kappa$-c.c. vs. $\kappa$-Knaster

Question

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say:

(a) $\mathbb{P}$ has the $\kappa$ chain condition ($\kappa$-c.c.) iff there is no subset of $\mathbb{P}$ of size $\kappa$ consisting of pairwise incompatible elements.

(b) $\mathbb{P}$ has the $\kappa$-Knaster property iff for every $A \subseteq \mathbb{P}$ of size $\kappa$ there is $B \subseteq A$ of size $\kappa$ consisting of pairwise compatible elements.

Clearly $\kappa$-Knaster implies $\kappa$-c.c. Also, it is trivial that the $\kappa$-c.c. implies the $\delta$-c.c. for every $\delta \geq \kappa$.

Question 1: If $\mathbb{P}$ is $\kappa$-Knaster, is it $\delta$-Knaster for every $\delta \geq \kappa$?

Question 2: If $\mathbb{P}$ is $\kappa$-c.c., is it necessarily $\kappa^+$-Knaster?

Answer 1

Partial answer: Clearly, if $\mathbb P$ is $\kappa$-c.c. and $\delta\to(\kappa,\delta)^2$, then $ \mathbb P$ is $\delta$-Knaster. Thus GCH answers Question 2 in the affirmative for every successor cardinal $\kappa$, because of the classical theorem: "if $2^{\aleph_n}=\aleph_{n+1}$, then $\aleph_{n+2}\to(\aleph_{n+1},\aleph_{n+2})^2$."

References:

P. Erdős, Some set-theoretical properties of graphs, Revista Universidad Nacional de Tucuman Serie A 3 (1942), 363-367.

P. Erdős and R. Rado, A problem on ordered sets, J. London Math. Soc. 28 (1953), 426-438; see p. 437.

P. Erdős and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489; see Theorem 4(iii) on p. 431, Theorem 7(i) on p. 432, Corollary 1 on p. 459.

Péter Komjáth and Vilmos Totik, Problems and Theorems in Classical Set Theory, Springer, 2006; see Problem 24.20 on p. 103, solution on pp. 412-413.

Answer 2

An answer to Question 1 (Thanks to hot_queen for the suggestion):

Consider the partial order $Fn(\aleph_\omega,2)$ of finite partial functions from $\aleph_\omega$ to 2, ordered by subset. Here is a large subset of $Fn(\aleph_\omega,2)$ without a large pairwise-compatible subset:

For each $\alpha \in [\aleph_0, \aleph_\omega)$, let $p_\alpha$ be a function with domain $[0,n] \cup \{ \alpha \}$, where $n$ is such that $\alpha \in [\aleph_n,\aleph_{n+1})$. For $\alpha \in [\aleph_n,\aleph_{n+1})$, let $p_\alpha(k) = 0$ for $k<n$, $p_\alpha(n) = 1$, and $p_\alpha(\alpha)=0$. There is no subset of size $\aleph_\omega$ of pairwise compatible elements, because if $\alpha < \beta$ are of distinct cardinalities, then their lower parts are disagree, and we must have such $\alpha, \beta$ in any $\aleph_\omega$-sized subset.