Hypergraphs with finitely many isomorphism subtypes

Let & = (H, E) be an n-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of 2? of cardinality X is finite for some infinite X . We solve a problem due independently to Jamison and Pouzet, by showing that there is a finite subset K of H such that the induced subgraph on H K is either empty or complete. We also characterize such hypergraphs in terms of finite (not necessarily uniform) hypergraphs. In a 1981 colloquium lecture at the University of South Carolina, R. Jamison posed the following problem: If an infinite «-uniform hypergraph H = (H, E) is isomorphic to each of its induced subgraphs of cardinality \H\ , must H be either empty (E = 0) or complete (E = [H]") ? In other words, must H be either independent or a clique? Recall that if tí c H, then the subgraph H\tí induced on tí is (tí ,Er\ [tí]"). In the case of graphs (n = 2) and 3-uniform hypergraphs of regular cardinality, there are simple affirmative solutions, found by Jamison and the authors and (independently) by M. Pouzet. But already the case of 3-uniform hypergraphs of singular cardinality is difficult enough so that it remained unsolved until 1985, shortly before the general solution provided in this paper; moreover, neither of the first two proofs for n = 3 by the authors seems to extend to n = 4. Our first theorem provides an affirmative answer to Jamison's problem under formally weaker hypotheses on H . Three of our other theorems also imply an affirmative answer. Given a hypergraph H, and induced subgraphs H, and H2 of H, we write H, ~ H2 if H, and H2 are isomorphic. This is obviously an equivalence relation, and if p < \H\ we let I (H) be the set of all isomorphism classes of induced subgraphs of cardinality ^, and let I.^, (H) be denoted 1(H). Jamison also posed the problem: if 1(H) is finite, must H either have a clique or an independent subset of cardinality \H\ ? (His first question is essentially Received by the editors February 3, 1986 and, in revised form, November 3, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 05C65, 05C75; Secondary 03C99. Research by H.A.K. supported in part by the National Science Foundation under ISP-80110451 and the Office of Naval Research under N00014-85K-0494; research by P.J.N. supported in part by the National Science Foundation under MCS-8301916. ©1989 American Mathematical Society 0002-9947/89 $1.00+ $.25 per page