If $\rho$ is a binary relation on a set $X$, the structure ${ \mathbb X}=\langle X,\rho\rangle$ is connected iff the minimal equivalence relation containing $\rho$ is the full relation on $X$. We show that, for a set $I$ the following conditions are equivalent (a) $|I|$ is less than the first measurable cardinal, (b) For each filter $\Phi \subset P(I)$ and each family $\{ { \mathbb X}_i :i\in I\}$ of binary structures, the reduced product $\prod_\Phi { \mathbb X}_i$ is connected, iff there are a finite set $K\subset I$ and $n\in \omega$ such that ${ \mathbb X}_i$ is connected, for each $i\in K$, and $\{ i\in I: { \mathbb X}_i \mbox{ is of diameter }\leq n\}\cup K\in \Phi$, (c)The ultraproduct $\prod_{ { \mathcal U}}{ \mathbb G}_{ \omega}$ is a disconnected graph for each non-principal ultrafilter ${ \mathcal U}\subset P(I)$, where ${ \mathbb G}_{ \omega}$ is the linear graph on $\omega$. Moreover, the implication "$\Leftarrow$" in (b) holds in ZFC.
Categories: math.LO