Set-Homogeneous Graphs

Uncountable Homogeneous Partial Orders

“…, then w < z (as otherwise w < g(y j ) for some j) and z ≤ w (as otherwise w > c β ). Thus, w < f β (X β ) and w A β \ f β (X β ), contrary to inductive assumption (3).…”

“…We mention one pair of results analogous to these. A graph is said to be k-sethomogeneous if any two isomorphic k-element induced subgraphs lie in the same orbit of the automorphism group (in its action on k-sets), and is (≤ k)-set-homogeneous if it is -set-homogeneous for all ≤ k. It is shown in [3,Section 4] that up to isomorphism there is up to complementation just one (≤ 3)-set-homogeneous countably infinite graph T which is not 2-homogeneous. However, by a construction in [3] completely different to that in the present paper, there are uncountable ≤ 3-set-homogeneous not 2-homogeneous graphs not elementarily equivalent to T or its complement (but they are classified up to elementary equivalence in [3]).…”

Minimality conditions on circularly ordered structures

“…Let Q * n be a structure Q n , K, L such that (i) its domain Q n is a countable dense subset of the unit circle, no two points making an angle of 2πk/n at the centre, where k ranges over integers, and (ii) for distinct x, y ∈ Q n , (x, y) ∈ σ i if and only if 2πi/n < arg(x/y) < 2π(i + 1)/n. By Proposition 2.1 of [14], Q * n is homogeneous and consequently Q * n admits elimination of quantifiers and is ℵ 0 -categorical. For each 0 ≤ i ≤ n − 1, and for any a ∈ Q n , σ i (a, Q n ) and σ i (Q n , a) are convex; hence Q * n is weakly circularly minimal.…”

Countable connected-homogeneous graphs

“…The problem of classifying the countable set-homogeneous graphs is open, but in [5] the countable set-homogeneous graphs that are not 3-homogeneous are described (in fact, up to complementation there is only one such graph).…”