The countable homogeneous poset Recognising R Peter J Cameron R is - PDF document

☎ ☛ ✄ ✝ ✞ ✄ ✂✄ ☛ ✂ ✂ ✂ ✂ ✍ The countable homogeneous poset Recognising R Peter J Cameron R is the unique countable graph with the property School of Mathematical Sciences that, for any finite graphs G and H with G H , every Queen Mary, University of London embedding of G in R extends to an embedding of H . London E1 4NS, U.K. p.j.cameron@qmul.ac.uk ✞ H ✞ G 1 : givn It is enough to require this when ✞✠✟ disjoint finite sets M 0 ✡ M 1 of vertices, there is a vertex Conference on Groups and Model Theory x joined to all vertices of M 1 and none of M 0 . Leeds, 11 April 2003 A similar characterisation holds for any countable homogeneous relational structure. Such a structure This is a commentary on the preprint is determined by its class of finite substructures, this “On homogeneous graphs and posets” ıss´ class having the amalgamation property ( Fra¨ e’s by Jan Hubiˇ cka and Jaroslav Neˇ setˇ ril Theorem ). Charles University, Prague, Czech Republic. 1 3 The random graph Constructions of R Erd˝ os–R´ enyi Theorem There is a unique countable graph R with the property that a random countable ✆ Vertex set is a countable model of set theory; x y graph X (obtained by choosing edges independently � 2 ) satisfies Prob if x y or y x . with probability 1 1 ✁ X R ✆ Vertex set is y if the x th binary digit of y is 1 ☞ ; x The graph R is (or vice versa ). ✆ universal : every finite or countable graph is ✆ Vertex set is the set of primes congruent to 1 embeddable in R . mod 4 ; x y if x is a quadratic residue mod y . ✆ homogeneous : any isomorphism between finite ✆ Vertex set is ✌ ; x y if the ✞ x y ✞ th term in a fixed subgraphs of R extends to an automorphism of R . universal binary sequence is 1 . These two properties characterise R . 2 4

☛ ✔ � ☛ ☛ ☛ ✓ ✍ ✄ ✂ ☛ ☞ ✔ ☛ ✟ � ☛ ✕ ✖ ✗ ✕ ✄ ✄ ✚ ✟ ✛ ✟ ✖ ✟ ☛ Motivation Lachlan–Woodrow Theorem The Erd˝ os–R´ enyi theorem is a non-constructive Lachlan and Woodrow determined all the countable existence proof for R . This can be re-formulated in homogeneous graphs. Apart from trivial cases, these 3 and their terms of Baire category instead of measure; in this are the Henson graphs H n for n form it applies to all countable homogeneous complements, and the random graph. H n is the relational structures. unique countable homogeneous K n -free graph which embeds all finite K n -free graphs, where K n is the complete graph on n vertices. However, an explicit construction can give us more information. Take a countable model of set theory. Let X be the 1 elements set of all sets which do not contain n For example, the fourth construction given earlier shows that R admits cyclic automorphisms (and, mutually comparable by the membership relation; put x y if x y or y x . This graph is isomorphic to H n . indeed, that the conjugacy classes of cyclic automorphisms of R are parametrised by the (This is essentially the same as Henson’s original construction of his graphs inside R .) universal binary sequences). 5 7 The generic digraph Models of set theory There is a digraph analogue of R (countable universal In showing that the first construction above gives R , homogeneous). Here is an explicit construction of it. we do not need all the axioms of ZFC: only the empty set, pairing, union, and foundation axioms. Take our model of hereditarily finite set theory. y if 2 x x if 2 x 1 Now put an arc x y , and y y . Sketch proof: Let M 0 and M 1 be disjoint finite sets. Let x M 1 ✎✑✏ y ✒ , where y is chosen so that it is not in 0 , take / Given M 0 ✡ M ✡ M with M 0 ✁ M ✎ M M 0 or in a member of a member of M 0 . (This ensures ✖✘☎ ✕ 1 that z x and x z for all z M 0 .) Then x is joined to ∑ 2 2 y ∑ 2 2 y 2 z x everything in M 0 and nothing in M 1 . y ✙ M y ✙ M where z is sufficiently large. Then there are arcs from elements of M to x , and from x to elements of M In particular, the Axiom of Infinity is not used. Now ✕ , but none between x and M 0 . there is a simple model of hereditarily finite set theory , satisfying the negation of the axiom of infinity: ☞ , and x y if the x th binary digit the ground set is If we restrict to the set of natural numbers for which of y is 1 . Thus the second construction is a special ✁ 2 i ✁ 2 i 1 ☎ st binary digits are not both 1 , the ☎ th and case of the first. for all i , we obtain the generic oriented graph. 6 8

✒ ✝ ✒ ✄ ✥ ✝ ✄ ☛ ☛ ✁ ✳ ✁ ✒ ☎ ✗ ✁ ✒ ✜ ✗ ✄ ✖ ✔ ☛ ✖ ✕ ✄ ✄ ✁ ✯ ✜ ✜ ✄ ☛ ☎ ☛ ✁ ☎ ☛ ✥ ✔ ✜ ✄ ✷ ☛ ☛ ☛ � ✷ ✢ ☛ ✷ ✄ ✷ ✳ ✳ ✷ ✒ ✏ ✄ ✱ ☎ ✄ ☎ ✄ ☎ ✄ ☛ ✄ ✁ ✱ ✄ ✁ ✒ ☛ ☎ ✗ ☛ Set theory with an atom The generic poset Take a countable model of set theory with a single The construction of the generic poset is similar to ✜ . Now let M be any set not containing atom ✜ . Putt that of the generic digraph just given. We restrict to a of the sets M not containing sub-collection M : M L ✏ A A ✒ , defined by the following recursive properties: ✜✣✒ : M R ✏ B B M ✒ . ✜✤☛ 0 ; / Correctness: M L ✎ M R and M L ✗ M R ✝✰✯ Then neither M L nor M R contains ✜ . Ordering: For all A M L and B M R , we have In the other direction, given two sets P ✡ Q whose 0 / ✏ A ✎ A R ✏ B ✎ B L elements don’t contain ✜ , let ☎✲✱ ✜✣✒ : B ✁ P ✞ Q P ✎✑✏ B Q ✁ P ✞ Q ✒ . Then ☎ doesn’t ✎✑✏ contain ✜ . Completeness: A L M L for all A M L , and B R M R for all B M R . Moreover, for any set M not containing ✜ , we have M ✁ M L ✞ M R ☎ . Now we put M N if Note that any set not containing can be ✏ M ✎ M R ✏ N ✎ N L 0 / ☎✴✱ represented in terms of sets not involving by means of the operation ✁✦✥✧✞★✥ Theorem The above-defined structure is isomorphic ✏ / 0 ✏ / 0 ✏ / 0 For example, ✜✩✒✪✒ is ✒★☎ . to the generic poset. ✒✫✞ ✡✦✏ 9 11 Part of the proof Note that / 0 ✁ / 0 ✞ / 0 ☎ is in ✯ ; the conditions are vacuously satisfied. The generic digraph again First, some notation. We define the level l ✁ M ☎ of an We define a directed graph as follows: ✁ / 0 0 and element M by the rules that l ☛✵✯ max ☎ : A 1 l ✁ M ✏ l ✁ A M L ✎ M R The vertices are the sets not containing ✜ . ✒✶✟ 0 . / for M If M ✡ N are vertices, then we put an arc N M if Also, if M N , then any element of N M L , and an arc M N if N M R . ✏ M ✎ M R ✏ N ✎ N L ☎ will be called a witness to M N . Note the following: Theorem This graph is the generic directed graph. (a) For M and A M L , B M R , we have For if M ✕ , M and M 0 are finite sets of vertices, with ☛✵✯ A M B . 0 , then we can find some z such / ✁ M ✎ M ✗ M 0 ✖✬☎ that x ✁ M ✎✑✏ z ✒✭✞ M ✕✮☎ has the correct arcs. (b) If W MN is a witness of M W MN N , then M N . (c) If W MN is a witness of M N , then either ✁ W MN ✁ W MN l l ✁ M ☎ or l l ✁ N ☎ . ☎✬✷ ☎✬✷ 10 12