Minimal Antichains in Well-founded Quasi-orders with an Application to Tournaments

We investigate the minimal antichains (in what is essentially Nash-Williamsʹ sense) in a well-founded quasi-order. We prove the following finiteness theorem: If Q is a well-founded quasi-order and k a fixed natural number, then there is a finite set Λk of minimal antichains of Q with the property that for any ideal I of Q obtained by excluding at most k elements of Q, I is well-quasi-ordered if and only if its intersection with each antichain in Λk is finite. When applied in a suitably sharpened form to an algorithmic problem arising in model theory, this yields a strengthening of the main result of [18].