Tilings in graphons

We introduce a counterpart to the notion of tilings, that is vertex-disjoint copies of a fixed graph F, to the setting of graphons. The case F=K2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional vertex covers and fractional matchings/tilings, and discuss connections with property testing. As an application of our theory, we determine the asymptotically almost sure F-tiling number of inhomogeneous random graphs G(n,W). As another application, in an accompanying paper (Hladký et al., 2019) we give a proof of a strengthening of a theorem of Komlós (Komlós, 2000).