Abstract. Let [Formula: see text] be the maximum number of copies of a graph [Formula: see text] in an [Formula: see text]-vertex graph which contains no copy of a graph from the graph family [Formula: see text]. We say that a rational number [Formula: see text] is realizable for [Formula: see text] if there exists a finite family [Formula: see text] such that [Formula: see text]. Using randomized algebraic constructions, Bukh and Conlon showed that every rational between 1 and 2 is realizable for [Formula: see text]. We generalize their result to show that every rational between 1 and [Formula: see text] is realizable for [Formula: see text] for all [Formula: see text]. We also determine the realizable rationals for stars and note the connection to a related Sidorenko-type supersaturation problem.

The game of plates and olives was originally formulated by Nicolaescu and encodes the evolution of the topology of the sublevel sets of Morse functions. We consider a random variant of this game. The process starts with an empty table. There are four different types of moves: (1) add a new plate to the table, (2) combine two plates and their olives onto one plate, removing the second plate from the table, (3) add an olive to a plate, and (4) remove an olive from a plate. We show that with high probability the number of olives is linear as the total number of moves goes to infinity. Furthermore, we prove that the number of olives is concentrated around its expectation.