Universal graphs with a forbidden near‐path or 2‐bouquet

We consider the problem of the existence of universal countable C ‐free graphs with C a connected finite graph. For C a tree arising by from a path by adjunction of one additional edge we show that a universal countable C ‐free graph exists. We determine precisely the 2‐bouquets C (i.e., unions of two complete graphs with jost one point in common) for which a universal countable C ‐free graph exists. We lay out some elements of a program for determining all the connected finite graphs C for which a countable universal C ‐free graph exists. One element of this program is the Tree Conjecture, which is now proved [ 2 ]. Our methods involve a mixture of model theory and combinatorics, with the Δ‐system lemma playing a significant role.