Set intersection representations for almost all graphs

Two variations of set intersection representation are investigated and upper and lower bounds on the minimum number of labels with which a graph may be represented are found which hold for almost all graphs. Speciically, if k (G) is deened to be the minimum number of labels with which G may be represented using the rule that two vertices are adjacent if and only if they share at least k labels, there exist positive constants c k and c 0 k such that almost every graph G on n vertices satisses c k n 2 log 2 n k (G) c 0 k n 2 log 2 n : Changing the representation only slightly by deening odd (G) to be the minimum number of labels with which G can be represented using the rule that two vertices are adjacent if and only if they share an odd number of labels results in quite diierent behaviour. Namely, almost every graph G satisses n ? p 2n ? dlog ne < odd (G) n ? 1: Furthermore, the upper bound on odd (G) holds for every graph.