Barely alternating real almost chains and extension operators for compact lines

Abstract

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 $E:C(K)⟶C(L)$ of norm at most three.