Sum-free sets and shift automorphisms Peter J. Cameron p.j.cameron@qmul.ac.uk Workshop on Graphs and Asynchronous Systems, 20 May 2008 Cayley graphs Theorem 1. There is a countable graph R such that, if a random graph X on a fixed countable vertex set is Let G be a group. A Cayley graph for G is a graph with vertex set G admitting G (acting by right mul- given by selecting edges independently with probability 1/2 , then Prob ( X ∼ tiplication) as a group of automorphisms. = R ) = 1 . Equivalently, it has edge set {{ g , sg } : g ∈ G , s ∈ S } , where S = S − 1 (to make it undirected) and Their proof was non-constructive, though ex- plicit constructions are known. I will give one be- 1 / ∈ S (to forbid loops). low. We denote this graph by Cay ( G , S ) . Sometimes it is assumed that S generates G Measure and category (equivalently, the graph is connected), but this is Two familiar techniques for non-constructive not necessarily the case here. existence proofs are: • Show that the set of all objects is a measure Shift graphs space, in which the “interesting” objects form These are Cayley graphs for the infinite cyclic a set of full measure. (Often the space has group Z . By abuse of notation, we let S denote the measure 1, and the argument can be phrased set of positive elements in the connection set, and in terms of probability. This is the case in the write Γ ( S ) for the graph Cay ( Z , S ∪ ( − S )) . Erd˝ os–R´ enyi theorem.) Thuus, x ∼ y in Γ ( S ) if and only if | x − y | ∈ S , • Show that the set of all objects is a complete where S ⊆ N . metric space, in which the interesting sets form a residual set, in the sense of Baire cate- The graph Γ ( S ) has a distinguished shift auto- gory (the complement of a set of the first cate- morphism, the map x �→ x + 1. gory – that is, a set which contains a countable It is easy to show that, if Γ ( S ) is isomorphic to intersection of open dense sets). Γ ( S ′ ) , then the two corresponding shift automor- In the Erd˝ os–R´ enyi Theorem, either measure or phisms of this graph are conjugate (in the auto- morphism group of Γ ) if and only if S = S ′ . category can be used: the graph R has measure 1 and is residual in the space of all graphs. The random graph A cautionary tale The following remarkable theorem was proved The set of all binary sequences is a probability by Erd˝ os and R´ enyi in 1963. space (recording the outcome of a sequence of coin 1
tosses) and a metric space (where the distance be- Theorem 4. Let X be a countable group which is not tween two sequences is 1/2 n if they first differ in the union of finitely many translates of non-principal the n th position). square-root sets. Then the set of Cayley graphs for X which are isomorphic to R is residual and has mea- By the Law of Large Numbers, almost all se- sure 1 . quences (in the sense of measure) have density 1/2. Many (but not all) countable groups satisfy this However, sequences with upper density 1 and condition. For example, in Z , any element has at lower density 0 form a residual set. most one square root. Measure and category do agree that almost all sequences are universal (see next slide). Countable homogeneous graphs A graph Γ is homogeneous if every isomorphism Universal sets between finite (induced) subgraphs of Γ extends to A binary sequence s is universal if every finite an automorphism of Γ . (An indced subgraph is a binary sequence σ occurs as a consecutive subse- subset of Γ in which both edges and nonedges are quence of s (i.e. there exists N such that s N + i = σ i the same as in Γ .) for i = 0, . . . , l ( σ ) − 1). The age of a graph Γ is the class of all finite The set of universal sequences has measure 1 graphs embeddable in Γ (as induced subgraphs). and is residual. A theorem of Fra¨ ıss´ e shows that there is at most A binary sequence is the characteristic function one countable homogeneous graph with any given of a subset S ⊆ N . We will say that the set S is age. universal if its characteristic function is universal. Fra¨ ıss´ e’s Theorem also gives a necessary and R as shift graph sufficient condition on a class to be the age of a countable homogeneous graph. The crucial con- Proposition 2. For S ⊆ N , the graph Γ ( S ) is isomor- dition is the amalgamation property : if two elements phic to R if and only if S is universal. of the age have isomorphic substructures, they can Ths shows that almost all shift graphs (in the be glued together along these substructures inside sense of either measure or category) are isomor- some structure in the age. phic to R . In addition, since sets of full measure or residual sets have cardinality 2 ℵ 0 , it follows: The theorem of Lachlan and Woodrow Corollary 3. The graph R has 2 ℵ 0 cyclic automor- Theorem 5. A countably infinite homogeneous graph phisms, pairwise not conjugate in Aut ( R ) . is one of the following: This also gives us an explicit construction of R , • a disjoint union of complete graphs of the same by taking an explicit universal set (for example, concatenate the base 2 representations of the natu- size; ral numbers). • complement of the preceding; R as Cayley graph • the unique countable homogeneous graph whose Thus a random Cayley graph for Z is almost age is the class of finite K n -free graphs for n ≥ 3 surely R . The same holds for a much wider class (this is the Henson graph H n ); of countable groups. • complement of the preceding; In a group X , a square-root set is a set of the form √ a = { x ∈ X : x 2 = a } ; • the random graph R. it is non-principal if a � = 1. The first two classes are not very interesting! 2
Cyclic automorphisms of Henson’s graphs Proposition 8. The probability that S consists entirely Henson showed that H 3 admits cyclic shifts but of odd numbers is non-zero (it is about 0.218 ). H n does not for n > 3. Conditioned on S consisting of odd numbers, it is almost surely of the form 2 S ′ + 1, where S ′ is uni- Is H 3 the random triangle-free Cayley graph for Z ? versal; that is, Γ ( S ) is almost surely the universal bipartite graph. Note that the Cayley graph Γ ( S ) is triangle-free if and only if S is sum-free , that is, x , y ∈ S ⇒ x + Why? y / ∈ S . For x , y , x + y ∈ S if and only if { 0, x , x + y } If we are constructing a random sum-free set S is a triangle in Γ ( S ) . and have no even numbers in a long initial seg- This leads us to the following definition: ment, then the odd numbers in the segment are random, and so the next even number has high Sum-free sets and sf-universal sets probability of being excluded; but the next odd A subset S of N is sf-universal if and only if number still has prbability 1/2 of being included. • S is sum-free; However, the pattern can change. For example, suppose that we chose 1 and 3 but not 5 or 7. Then • for any finite binary sequence σ , either we might choose 8 and 10, and the event that all subsequent numbers are congruent to 1, 3, 8 or 10 – there exist i < j with σ i = σ j = 1 and mod 11 has positive probability. j − i ∈ S ; or – σ occurs as a consecutive subsequence of Other events with positive measure the characteristic function of S . A subset T of Z / ( n ) is complete sum-free if it is sum-free, and if for any z / ∈ T there exist x , y ∈ T In other words, S is a sum-free set in which ev- such that z = x + y . For example, { 2, 3 } mod 5 is ery subsequence not forbidden by the sum-free complete sum-free; so is the set { 1, 3, 8, 10 } mod 11 condition actually occurs somewhere. we saw on the last slide. sf-universal sets and Henson’s graph Proposition 9. The probability that S is contained in the set of congruence classes corresponding to a fixed Proposition 6. Γ ( S ) ∼ = H 3 if and only if S is sf- complete sum-free set mod n is strictly positive. universal. Proposition 10. Prob ( 2 is the only even number in S ) > Proposition 7. The sf-universal sets are residual in the 0 . class of sum-free sets. The last two results have a common generalisa- So H 3 is the generic cyclic triangle-free graph in tion. The class of sum-free sets which fall into a the sense of Baire category. complete sum-free set mod n after some point also What about measure? has positive probability. Random sum-free sets What else? There is a simple measure for sum-free sets: But it is unlikely that we have yet caught almost all sum-free sets! Consider the natural numbers in turn. Conjecture 1. Prob ( S is sf-universal ) = 0 . When considering n , if n = x + y where x , y ∈ S , then n / ∈ S ; otherwise toss a fair It is not feasible to go on finding classes with coin to decide. positive probability and adding up the probabil- ities until we get everything! The probability of The first surprise is that we do not obtain an sf- getting a set of odd numbers is only known to universal set almost surely: three decimal places. 3
Recommend
More recommend
