Barely alternating real almost chains and extension operators for compact lines

Abstract

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