Barely alternating real almost chains and extension operators for compact lines

Assume 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∈ω 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) C(L) of norm at most three.