Quasi-randomness of graph balanced cut properties

Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random graph of the same edge density. Recently, Shapira and Yuster proved the following result on quasi-randomness of graphs. Let k ≥ 2 be a fixed integer, α1, . . . , αk be positive reals satisfying ∑ i αi = 1 and (α1, . . . , αk) 6= (1/k, . . . , 1/k), and G be a graph on n vertices. If for every partition of the vertices of G into sets V1, . . . , Vk of size α1n, . . . , αkn, the number of complete graphs on k vertices which have exactly one vertex in each of these sets is similar to what we would expect in a random graph, then the graph is quasi-random. However, the method of quasi-random hypergraphs they used did not provide enough information to resolve the case (1/k, . . . , 1/k) for graphs. In their work, Shapira and Yuster asked whether this case also forces the graph to be quasi-random. Janson also posed the same question in his study of quasi-randomness under the framework of graph limits. In this paper, we positively answer their question.