Considering different structures (graphs, numbers, permutations, etc.) this questions has different aspects and we may have several nonequivalent answers. In the talk we shall concentrate on two topics.
(a) Quasi-random graphs Thomason and Chung - Graham - Wilson gave a class of graph properties, all possessed by random graphs and at the same time equivalent to each other. With Simonovits we proved, that this class (quasi-random graphs) can be characterized also in the Szemeredi-partition of graphs. Using this approach we proceed, that some properties, which do not imply quasi-randomness on their own, do imply it if we consider the corresponding hereditarily extended properties. (Large subgraphs of random-like graphs must also be random-like.)
(b) Quasi-periodic sequences; the distribution of ({nx}), ([nx]) sequences, for x irrational. ({} denotes "fractional part", [] denotes "integer part".) We remark, that ({nx}), ([nx]) sequences can be considered as the one-dimensional analog of Penrose-tilings (related to the atomic structures of quasi-crystals, which are ``between'' crystal and amorphous atomic structures.)
The results may contribute to study the problem: how to create a scale between periodic and random sequences, between non-random and random graphs, etc.