A New Construction Principle

We use the framework of Abstract Elementary Classes (AECs) to introduce a new Construction Principle CP(K,), which strictly generalises the Construction Principle of Eklof, Mekler and Shelah and allows for many novel applications beyond the setting of universal algebra. In particular, we show that CP(K,) holds in the classes of free products of cyclic groups of fixed order, direct sums of a fixed torsion-free abelian group of rank 1 which is not Q, (infinite) free $(k,n)$-Steiner systems, and (infinite) free generalised n-gons. From this we derive, in ZFC, that these classes of structures are not axiomatisable in the logic L∞,ω₁, and, under $V=L$, that they are not axiomatisable in L∞,∞.