I ran into a claim concerning Woodin's fast function forcing in the following paper of Apter and Cummings which sounds no right to me:

A. Apter, J. Cummings, Blowing up the power set of the least measurable, Journal of Symbolic Logic 67 (2002), no. 3, 915--923.

**Definition:** Woodin's fast function forcing on $\kappa$ consists of partial functions $p$ from $\kappa$ to $\kappa$ ordered by inclusion such that:

The domain of $p$ consists of inaccessible cardinals $\lambda <\kappa$ which are closed under $p$, namely for every $\lambda, \theta\in dom(p)$ if $\theta<\lambda$ then $p(\theta)<\lambda$.

For every $\lambda\in dom(p)$ we have $|dom(p)\cap \lambda|<\lambda$

The claim is as follows:

**Claim:** For Mahlo $\kappa$ the fast function forcing satisfies $\kappa$-Knaster property.

I have seen no such claims discussing the chain condition of fast function forcing in other related papers.

Also I think the following counterexample works for refuting the $\kappa$-cc property of the fast function forcing as defined above.

.**Counterexample:** Fix an inaccessible cardinal $\lambda <\kappa $. Consider the set of $\kappa$-many incompatible conditons as follows: $p_{\alpha}:=\{\langle\lambda,\alpha\rangle\}$ for $\alpha<\kappa$.

Finally the questions are:

Question 1:Am I missing something?! Does fast function forcing really have $\kappa$-Knaster property?

Question 2:If $f$ is the fast function added by fast function forcing, is the following statement true?For all functions $g:\kappa\rightarrow\kappa$ in the forcing extension there is a function $h:\kappa\rightarrow\kappa$ such that $\forall \alpha\in \kappa ~~~ g(\alpha)<f(h(\alpha))$.