On set intersection representations of graphs

The intersection dimension of a bipartite graph with respect to a type L is the smallest number t for which it is possible to assign sets Ax ⊆ {1, . . . , t} of labels to vertices x so that any two vertices x and y from different parts are adjacent if and only if |Ax ∩Ay| ∈ L. The weight of such a representation is the sum ∑x |Ax| over all vertices x. We exhibit explicit bipartite n×n graphs whose intersection dimension is: (i) at least n1/|L| with respect to any type L, (ii) at least √ n with respect to any type of the form L = {k,k + 1, . . .}, and (iii) at least n1/|R| with respect to any type of the form L = {k | k mod p ∈ R}, where p is a prime number. We also show that any intersection representation of a Hadamard graph must have weight about n lnn/ ln lnn, independent on the used type L. Finally, we formulate several problems about intersection dimensions of graphs related to some basic open problems in the complexity of boolean functions.