Saturation of null ideal In ZFC, can we find more than continuum many non null sets of reals whose pairwise intersections are null?
 A: The answer is no. Observe that it is enough to show that, consistently, the density of the boolean algebra $\mathcal{P}(\mathbb{R}) / Null$ is continuum (which means that there are continuum many non null sets such that every non null set contains one of them). For this, it suffices to show that, consistently, ($\star$) holds:
($\star$): $2^{\omega} = \kappa = \kappa^{< \kappa}$ and every non null set of reals has a non null subset of size less than $\kappa$.
Claim 1: If $\kappa = 2^{\omega}$ is real valued measurable, then ($\star$) holds.
Proof: That $\kappa^{< \kappa} = \kappa$, is a result of Prikry (Theorem 22.2 in Jech's book). Let $\langle x_i : i < \kappa \rangle$ be a one-one enumeration of a non null set of reals $X$. Force with the null ideal of a measure on $\mathcal{P}(\kappa)$. Let $j: V \to M$ be the generic elementary embedding with critical point $\kappa$. Note that, in $M$, $X$ is a non null initial segment of $j(X)$ - this is because forcing with a measure algebra preserves old non null sets; so this also holds in $V$.
Claim 2: ($\star$) holds in the random real model.
Proof: (Communicated by Arnold Miller) Let $V \models GCH$, and $P$ add $\omega_2$ random reals $\langle r_i : i < \omega_2 \rangle$. Suppose $X = \langle x_i : i < \omega_2 \rangle$ is non null. Define $X_i = V[\langle r_j : j < i \rangle] \cap X$. Choose $\alpha < \omega_2$ of cofinality $\omega_1$ such that no null set coded in $V[\langle r_j : j < \alpha \rangle]$ covers $X_{\alpha}$. So $X_{\alpha}$ is non null in $V[\langle r_j : j < \alpha \rangle]$ and it remains so in $V[\langle r_i : i < \omega_2 \rangle]$.
