Barely alternating real almost chains and extension operators for compact lines

Assume \(\text {MA}(\kappa )\) $\text{MA}(\kappa )$ . We show that for every real chain of size \(\kappa \) $\kappa$ in the quotient Boolean algebra \(P(\omega )/fin\) $P(\omega )/fin$ we can find an almost chain of representatives such that every \(n\in \omega \) $n\in \omega$ oscillates at most three times along the almost chain. This is used to show that for every countable discrete extension of a separable compact line K of weight \(\kappa \) $\kappa$ there exists an extension operator \(E:C(K)\longrightarrow C(L)\) $E:C(K)⟶C(L)$ of norm at most three.