Barely alternating real almost chains and extension operators for compact lines

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