Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy^-1 ≠ zw^-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn³. For elementary Abelian groups of square order, ❘G❘ ⩾ n² is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with <cn²/log²n vertices. We comment on embedding trees and, in particular, stars, as induced subgraphs of Cayley graphs, and on the related problem of product-free (sum-free) sets in groups. We summarize the known results on the cardinality of Sidon sets of infinite groups, and formulate a number of open problems.

We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate.