Barely alternating real almost chains and extension operators for compact lines

Assume \(\text {MA}(\kappa )\). We show that for every real chain of size \(\kappa \) in the quotient Boolean algebra \(P(\omega )/fin\) we can find an almost chain of representatives such that every \(n\in \omega \) 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 \(\kappa \) there exists an extension operator \(E:C(K)\longrightarrow C(L)\) of norm at most three.