SMALL UNIVERSAL FAMILIES OF GRAPHS ON ℵ ω+1 JAMES CUMMINGS, MIRNA D ˇ ZAMONJA, AND CHARLES MORGAN Abstract. We prove that it is consistent that ℵ ω is strong limit, 2 ℵω is large and the universality number for graphs on ℵ ω+1 is small. The proof uses Prikry forcing with interleaved collapsing. 1. Introduction If μ is an infinite cardinal, a universal graph on μ is a graph with vertex set μ which contains an isomorphic induced copy of every such graph. More generally, a family F of graphs on μ is jointly universal if every graph on μ is isomorphic to an induced subgraph of some graph in F . We denote by u μ the least size of a jointly universal family of graphs on μ, and record the easy remarks that u μ ≤ 2 μ and that if u μ ≤ μ then u μ = 1. If μ = μ <μ , then by standard results in model theory there exists a saturated (and hence universal) graph on μ. It follows that under GCH and the hypothesis that μ is regular, u μ = 1. A standard idea in model theory (the construction of special models) shows that under GCH we have u μ = 1 for singular μ as well: we fix hμ i : i< cf(μ)i a sequence of regular cardinals which is cofinal in μ, build a graph G which is the union of an increasing sequence of induced subgraphs G i where G i is a saturated graph on μ i , and argue by repeated applications of saturation that G is universal. Questions about the value of u μ when μ<μ <μ have been investigated by several authors. We refer the reader to papers by Dˇ zamonja and Shelah [4, 3], Kojman and Shelah [6], Mekler [10] and Shelah [13]. We will consider the case when μ is a successor cardinal κ + and 2 κ >κ + . When κ is regular it is known that: James Cummings was partially supported by NSF grant DMS-1101156. Mirna Dˇ zamonja thanks EPSRC for their support through their grant EP/I00498 and Leverhulme Trust for a Research Fellowship for the period May 2014 to May 2015. Charles Morgan thanks EPSRC for their support through grant EP/I00498. Cum- mings, Dˇ zamonja and Morgan thank the Institut Henri Poincar´ e for their support through the “Research in Paris” program during the period 24-29 June 2013. The authors thank Jacob Davis for his useful comments on draft versions of this paper. 1 arXiv:1408.4188v1 [math.LO] 19 Aug 2014