Is Ramseys theorem omega-automatic?

Symposium on Theoretical Aspects of Computer Science 2010 (Nancy , Fr ance), pp. 537-548 www .st acs-conf .org IS RAMSEY’S THEOREM ω -A UTOMA TIC? DIETRICH KUSKE Cen tre national de la rec herche scien tifique (CNRS) and Lab ora toire Bordelais de Recherc he en Informatique (LaBRI), Bordeaux, F rance Abstra ct. W e study the existence of infi nite cliques in ω -automatic (hyper- )graphs. It turns out that the situation is muc h nicer than in general u ncoun table graphs, but not as nice as for automatic graphs. More sp ecifically , we sh o w that every u ncoun table ω -automatic graph contains an un- countable co-context-free clique or anticlique, b ut n ot necessarily a con text-free (let alone regular) clique or anticlique. W e also sho w that un coun table ω -aut oma tic ternary hyp er- graphs need n ot hav e u ncoun table cliques or anticliques at all. In tro duction Ev ery infinite graph h as an in finite clique or an infin ite antic lique – this is the paradig- matic formulati on of Ramsey’s theorem [Ram30]. But this theorem is highly non-construc- tiv e since there a re recursive infinite graphs whose in finite cliques and an ticliques are all non-recursive (not ev en in Σ 0 2 , [Jo c72], cf. [Gas98, Thm. 4.6]). Recall that a graph is re- cursiv e if b oth its set of no des and its set of edges can b e d ecided by a T urin g mac hine. Replacing these T uring mac hines b y fin ite automata, one obtains the more restrictiv e no- tion of an automatic gr aph : the set of no des is a regular set and w h ether a pair of no des forms an ed ge can b e decided by a sync hronous tw o- tap e automaton (this concept is kno wn since th e b eginning of automata theory , a systematic study started with [KN95, BG04], s ee [Rub08] for a recen t ov erview). In this con text, the situ ation is muc h more f av o urable: ev ery infinite automatic graph con tains an infi nite regular clique or an infin ite regular anti clique (cf. [Rub08]). So on after Ramsey’s pap er f rom 1930, authors got int erested in a quan titativ e analysis. F or fi nite graphs, one can ask for th e minimal n umber of no des that guarantee the existence of a clique or antic lique of some pr escribed size. This also m akes sense in the infi nite: ho w man y n odes are n ec essary an d sufficient to obtain a clique or anti clique of size ℵ 0 (Ramsey’s theorem tells us: ℵ 0 ) or ℵ 1 (here one n eeds more than 2 ℵ 0 no des [Sie33, ER56]). Since automatic graph s con tain at most ℵ 0 no des, we need a more general n ot ion for a r ecur sion-theo retic analysis of this situation. F or this, we use Blumensath & Gr¨ adel’s [BG04] ω -automatic graphs: the names of n odes form a regular ω -language and the edge 1998 ACM Subje ct Cl ass ific ation: F.4.1. Key wor ds and phr ases : Logic in computer science, Automata, Ramsey theory . These results were obtained when the aut hor was affiliated with the Universit¨ at Leipzig. c  Dietrich Kusk e CC  Creative Commons Attribution-NoDer ivs License 538 DIETRICH KUSKE relation (on names) as well as the r ela tion “these t w o names denote the same n ode” can b e decided by a synchronous 2-tap e B ¨ uc hi-automaton. In this pap er, we answ er th e question whether these ω -automatic graphs are m ore like automatic graphs (i.e., large cliques or an ticliques with nice prop erties exist) or lik e general graphs (large cliques need not exist). Our answe r to th is question is a clear “somewhere in b et w een”: W e show th at ev ery ω -automatic graph of size 2 ℵ 0 con tains a clique or antic lique of size 2 ℵ 0 (Theorem 3.1) – this is in contrast to the case of arbitrary graphs where s uc h a sub graph need not exist [Sie33]. But in general, there is n o regular clique or anti clique (Theorem 3.13) – this is in con trast with the case of automatic graphs where we alw a y s fi nd a large regular clique or an ticlique. Finally , w e also p r o vide an ω -automatic “ternary h yp ergraph” of size 2 ℵ 0 without any clique or anticl ique of size ℵ 1 , let alone 2 ℵ 0 (Theorem 3.11). F or Theorem 3.1, w e r e-us e the p roof from [BKR08] that was originally constructed to deal with infin it y quantifiers in ω -automatic stru ctur es. The pro of of Theorem 3.13 makes use of the “ultimately equal” relation. This relation wa s also crucial in th e separation of injectiv ely from general ω -automatic structures [HKMN08] as we ll as in the handling of infinity quantifiers in [KL08] and [BKR08]. In the ternary hypergraph from Th eo rem 3.11, a 3-set { x, y , z } of infi nite words with x < lex y < lex z forms an undir ec ted h yp eredge iff the longest common prefix of x an d y is sh orter than the longest common p refix of y and z . ¿F rom Theorem 3.1 (i.e., the existence of large cliques or an ticliques in ω -automatic graphs), we deriv e that any ω -automatic partial order of size 2 ℵ 0 con tains an an tichain of size 2 ℵ 0 or a copy of th e real line. 1. Prelimina ries 1.1. Ramsey-theory F or a set V and a natural num b er k ≥ 1, let [ V ] k denote the set of k -eleme nt subsets of V . A ( k , ℓ ) -p artition is a p air G = ( V , E 1 , . . . , E ℓ ) where V is a set and ( E 1 , . . . , E ℓ ) is a p artitio n of [ V ] k in to (p ossibly empt y) sets. F or 1 ≤ i ≤ ℓ , a set W ⊆ V is E i - homo gene ous if [ W ] k ⊆ E i ; it is homo gene ous if it is E i -homogeneous for some 1 ≤ i ≤ ℓ . The case k = ℓ = 2 is sp ecial: any (2 , 2)-partition G = ( V , E 1 , E 2 ) can b e considered as an (undirected lo op-free) graph ( V , E 1 ). Homogeneous sets in G are th en complete or d iscrete induced s ubgraphs of ( V , E 1 ). Ramsey th eo ry is concerned with th e follo win g question: Do es ev ery ( k , ℓ )-partition G = ( V , E 1 , . . . , E ℓ ) with | V | = κ hav e a h omo geneous set of s iz e λ (wh er e κ and λ are cardinal num b ers and k , ℓ ≥ 2 are natur al num b ers). If th is is the case, on e writes κ → ( λ ) k ℓ (a notation due to Erd˝ os and Rado [ER56]). T his allo ws to form ulate Ramsey’s theorem concisely: Theorem 1.1 (Ramsey [Ram30 ]) . If k , ℓ ≥ 2 , then ℵ 0 → ( ℵ 0 ) k ℓ . In particular, every graph with ℵ 0 no des cont ains a complete or discrete ind uced su b- graph of the same size. If one wan ts to find h omog eneous sets of size ℵ 1 , the base set has to b e muc h larger: Theorem 1.2 (Sierpi ´ nski [Sie33]) . If k , ℓ ≥ 2 , then 2 ℵ 0 6→ ( ℵ 1 ) k ℓ and ther efor e in p articular 2 ℵ 0 6→ (2 ℵ 0 ) k ℓ . IS RAMSEY’S THEOREM ω -AUTOMA TIC? 539 Erd˝ os an d Rado [ER56] pr o v ed that partitions of size pr operly larger than 2 ℵ 0 ha v e homogeneous sets of size ℵ 1 . F or more details on infin ite Ramsey theory , see [Jec02, Chap- ter 9]. 1.2. ω -languages Let Γ b e a fi nite alphab et. With Γ ∗ w e den ote the set of all finite words o ver the alphab et Γ. The set of all n onempt y fin ite w ord s is Γ + . An ω -wor d o v er Γ is an infinite ω -sequence x = a 0 a 1 a 2 · · · with a i ∈ Γ, w e set x [ i, j ) = a i a i +1 . . . a j − 1 for natural num b ers i ≤ j . In the same s pirit, x [ i, ω ) d en ot es the ω -w ord a i a i +1 . . . . The set of all ω -w ords o ver Γ is d enote d by Γ ω and Γ ∞ = Γ ∗ ∪ Γ ω . F or a set V ⊆ Γ + of finite words let V ω ⊆ Γ ω b e the set of all ω -wo rds of the form v 0 v 1 v 2 · · · with v i ∈ V . Tw o infinite w ords x, y ∈ Γ ω are ultimately e qual , b riefly x ∼ e y , if ther e exists i ∈ N with x [ i, ω ) = y [ i, ω ). By ≤ lex , we denote the lexicographic ord er on the set Σ ω (with some, imp licitly assumed linear order on th e letters f rom Σ) and ≤ pref the prefix order on Σ ∞ . F or Σ = { 0 , 1 } , the supp ort su pp( x ) ⊆ N is the set of p ositions of the letter 1 in th e w ord x ∈ Σ ω . A (nondeterministic) B ¨ uchi-automa ton M is a tu ple M = ( Q, Γ , δ, ι, F ) w here Q is a finite set of states, ι ∈ Q is the initial state, F ⊆ Q is the set of final states, and δ ⊆ Q × Γ × Q is the transition relation. If Γ = Σ n for some alphab et Σ, then we sp eak of an n - dimensio nal B¨ uchi-automaton over Σ. A run of M on an ω -w ord x = a 0 a 1 a 2 · · · is an ω -w ord r = p 0 p 1 p 2 · · · o ver the set of states Q such th at ( p i , a i , p i +1 ) ∈ δ for all i ≥ 0. The run r is suc c e ssfu l if p 0 = ι and there exists a fi nal state from F that o ccurs infi nitely often in r . The ω -language L ( M ) ⊆ Γ ω defined by M is the set of all ω -w ords that ad m it a su cc essful r un. An ω -language L ⊆ Γ ω is r e gular if there exists a B ¨ uc hi-automaton M with L ( M ) = L . Alternativ ely , regular ω -languages can b e represented algebraically . T o this end, one defines ω -semigr oups to b e t w o-sorted algebras S = ( S + , S ω ; · , ∗ , π ) where · : S + × S + → S + and ∗ : S + × S ω → S ω are binary op erations and π : ( S + ) ω → S ω is an ω -ary op eration such that the follo win g hold: • ( S + , · ) is a semigroup, • s ∗ ( t ∗ u ) = ( s · t ) ∗ u , • s 0 · π (( s i ) i ≥ 1 ) = π (( s i ) i ≥ 0 ), • π (( s 1 i · s 2 i · · · s k i i ) i ≥ 0 ) = π (( t j ) j ≥ 0 ) wheneve r ( t j ) j ≥ 0 = ( s 1 0 , s 2 0 , . . . , s k 0 0 , s 1 1 , . . . , s k 1 1 , . . . ) . The ω -semigroup S is finite if b oth, S + and S ω are fin ite. The free ω -semigroup generated b y Γ is Γ ∞ = (Γ + , Γ ω ; · , ∗ , π ) where u · v and u ∗ x are the natural op erations of prefi x in g a w ord by the finite w ord u , and π (( u i ) i ≥ 0 ) is the omega-w ord u 0 u 1 u 2 . . . . A homomorph ism h : Γ ∞ → S of ω -semigroups maps fi nite words to elemen ts of S + and ω -words to elements of S ω and commutes with the op erations · , ∗ , and π . The algebraic c haracterisation of r egula r ω -languages then reads as follo ws. Prop osition 1.3. An ω -language L ⊆ Γ ω is r e gular if and only if ther e e xists a finite ω -semigr oup S , a set T ⊆ S ω , and a homomorphism η : Γ ∞ → S such that L = η − 1 ( T ) . 540 DIETRICH KUSKE Hence, ev ery B ¨ uc h i-automaton is “equiv alen t” to a homomorphism into some fin ite ω -semigroup together with a distinguished set T (and vice versa). F or ω -w ord s x i = a 0 i a 1 i a 2 i · · · ∈ Γ ω , th e c onvolution x 1 ⊗ x 2 ⊗ · · · ⊗ x n ∈ (Γ n ) ω is defined b y ( x 1 , . . . , x n ) ⊗ = ( a 0 1 , . . . , x 0 n ) ( a 1 1 , . . . , a 1 n ) ( a 2 1 , . . . , a 2 n ) · · · . An n -ary relation R ⊆ (Γ ω ) n is called ω -automatic if the ω -language { ( x 1 , . . . , x n ) ⊗ | ( x 1 , . . . , x n ) ∈ R } is regular. T o d escribe the complexit y of ω -languages, w e will use language-theoretic terms. L et LANG denote the class of all languages (i.e., sets of finite w ords o v er some finite set of sy m - b ols) and ω LANG the class of all ω -languages. By REG and ω REG, we denote the regular languages and ω -languages, r esp. An ω -language is c ontext-fr e e if it can b e accepted b y a pushd o wn-automaton with B ¨ uc hi-acceptance (on states), it is c o-c ontext-fr e e if its comple- men t is con text-free. W e denote by ω CF the set of conte xt-free ω -languages and by co- ω CF their complemen ts. An ω -language b elongs to LANG ∗ if it is of the form S 1 ≤ i ≤ n U i V ω i with U i , V i ∈ LANG. Then ω REG ⊆ LANG ∗ and ω C F ⊆ LANG ∗ where th e sets U i and V i are regular and con text-free, resp [Sta97]. In b et w een these t wo classes, w e defi ne the class ω erCF of eventual ly r e gular c ontext-fr e e ω -language s that comprises all sets of the form S 1 ≤ i ≤ n U i V ω i with U i ∈ LANG con text-free and V i ∈ LANG regular. Alternativ ely , ev entually regular con text-free ω -languages are the finite unions of ω -languages of the form C · L wher e C is a con text free-language and L a r eg ular ω -language. Let co- ω erCF denote the set of complemen ts of ev en tually regular conte xt-free ω -languages. A final, rather p eculiar class of ω -languages is Λ : it is the class of ω -languages L suc h that ( R , ≤ ) em b eds in to ( L, ≤ lex ) (the name d er ives from the notation λ for th e order t yp e of ( R , ≤ )). 1.3. ω -aut omat ic ( k , ℓ ) -part itions An ω - auto matic pr esentation of a ( k , ℓ ) -p artition ( V , E 1 , . . . , E ℓ ) is a pair ( L, h ) consist- ing of a regular ω -language L and a surjection h : L → V suc h th at { ( x 1 , x 2 , . . . , x k ) ∈ L k | { h ( x 1 ) , h ( x 2 ) , . . . , h ( x k ) } ∈ E i } for 1 ≤ i ≤ k an d R ≈ = { ( x 1 , x 2 ) ∈ L 2 | h ( x 1 ) = h ( x 2 ) } are ω -automatic . An ω -automatic p r esen tation is inje ctive if h is a bijection. A ( k , ℓ )-partition is (inje ctiv ely) ω -automatic if it has an (injectiv e) ω -automatic pr esen tation. F rom [BKR08], it follo ws that an uncountable ω -automatic ( k , ℓ )-partition h as 2 ℵ 0 elemen ts. This pap er is concerned with the qu esti on w hether every (injectiv e) ω -automatic p re- sen tation ( L, h ) of a ( k , ℓ )-partition admits a “simple” set H ⊆ L such that h ( H ) has λ elemen ts and is homogeneous. More precisely , let C b e a class of ω -languages, k , ℓ ≥ 2 natural num b ers, and κ and λ cardinal num b ers. Then we write ( κ, ω A ) → ( λ, C ) k ℓ if the follo win g partition prop ert y holds: for ev ery ω -automatic pr esen tation ( L, h ) of a ( k , ℓ )-partition G of size κ , there exists H ⊆ L in C such that h ( H ) is homogeneous in G and of size λ . ( κ, ω iA ) → ( λ, C ) k ℓ is to b e und erstoo d similarly wh ere w e only consider injectiv e ω -automatic pr esentati ons. Remark 1.4. Let G = ( V , E 1 , . . . , E ℓ ) b e s ome ( k , ℓ )-partition with ω -automatic present a- tion ( L, h ). Then the partition pr op erty ab o ve requires that there is a “large” homogeneous set X ⊆ V a nd an ω -language H ∈ C s uc h th at h ( H ) = X , in particular, ev ery elemen t of IS RAMSEY’S THEOREM ω -AUTOMA TIC? 541 X has at least one r epresen tativ e in H . Alternativ ely , one could require that h − 1 ( X ) ⊆ L is an ω -language from C . In this pap er, we only encoun ter classes C of ω -languages suc h that the follo win g closur e prop ert y h olds: if H ∈ C and R is an ω -automatic relation, then also R ( H ) = { y | ∃ x ∈ H : ( x, y ) ∈ R } ∈ C . S ince h − 1 h ( H ) = R ≈ ( H ), all ou r r esults also hold for th is alternativ e requ ir emen t h − 1 ( X ) ∈ C . This pap er shows (0) if k , ℓ ≥ 2, then ( ℵ 0 , ω A ) → ( ℵ 0 , ω REG) k ℓ , b u t (2 ℵ 0 , ω A ) 6→ ( ℵ 0 , ω REG) k ℓ , see Theo- rem 2.1. (1) if ℓ ≥ 2, then (2 ℵ 0 , ω A ) → (2 ℵ 0 , co- ω er C F) 2 ℓ , see Theorem 3.1. (2) if k ≥ 3, ℓ ≥ 2, and λ > ℵ 0 , then (2 ℵ 0 , ω iA ) 6→ ( λ, ω LANG) k ℓ , see Theorem 3.11. (3) if k , ℓ ≥ 2 and λ > ℵ 0 , then (2 ℵ 0 , ω iA ) 6→ ( λ, ω CF ) k ℓ , see Th eorem 3.13. Here, the fir st p art of (0) is a strengthening of R amsey’s theorem since the infinite homo- geneous set is regular. The second part might lo ok surpr ising since larger ( k , ℓ )-partitions should hav e larger homogeneous sets – b ut not n ece ssarily regular ones! In con trast to Sierpi ´ nski’s resu lt, (1) sho ws that ω -automatic (2 , ℓ )-partitions ha v e a larger d eg ree of ho- mogeneit y than arbitrary (2 , ℓ )-partitions. Ev en more, the complexity of the homogeneous set can b e b ound in language-theoreti c terms (there is alw ays a homogeneous s et that is the complement of an ev entuall y regular con text-free ω -language). Statement (2) is an analogue of S ierpi´ n ski’s Theorem 1.2 sho wing that (injectiv e) ω -automatic ( k , ℓ )-partitions are as in-homogeneous as arbitrary ( k , ℓ )-partitions p ro vided k ≥ 3. T h e complexity b ound from (1) is shown to b e optimal b y (3) p r o ving that one cannot alw ays find context- free homogeneous sets. Hence, d esp ite the existence of large homogeneous sets for k = 2, for some ω -automatic presenta tions, they are b ound to ha v e a certain (lo w) level of complexity that is h igher than the regular ω -languages. 2. Coun tably infinite homogeneous sets Let k , ℓ ≥ 2 b e arbitrary . Then, fr om Ramsey’s theorem, w e obtain imm ediat ely ( ℵ 0 , ω A ) → ( ℵ 0 , ω LANG) k ℓ and (2 ℵ 0 , ω A ) → ( ℵ 0 , ω LANG) k ℓ , i.e., all infi nite ω -automatic ( k , ℓ )-partitions h av e homogeneous s ets of size ℵ 0 . In this section, we ask w hether suc h homogeneous sets can alw a ys b e chosen regular: Theorem 2.1. L et k , ℓ ≥ 2 . Then (a) ( ℵ 0 , ω A ) → ( ℵ 0 , ω REG) k ℓ . (b) (2 ℵ 0 , ω iA ) → ( ℵ 0 , ω REG) k ℓ . (c) (2 ℵ 0 , ω A ) 6→ ( ℵ 0 , LANG ∗ ) k ℓ , and ther efor e in p articular (2 ℵ 0 , ω A ) 6→ ( ℵ 0 , ω CF) k ℓ and (2 ℵ 0 , ω A ) 6→ ( ℵ 0 , ω REG ) k ℓ . Pr o of. Let ( L, h ) b e an ω -automatic presen tation of some ( k, ℓ )-partition G = ( V , E 1 , . . . , E ℓ ) with | V | = ℵ 0 . By [BKR08], there exists L ′ ⊆ L regular suc h that ( L ′ , h ) is an in jecti v e ω -automatic pr esen tation of G . F rom a B ¨ uc hi-automaton for L ′ , one can compu te a fi n ite automaton accepting some language K su c h that ( K, h ′ ) is an injectiv e automatic p r esen ta- tion of G [Blu99 ]. Hence, b y [Rub 08 ], there exists a regular set H ′ ⊆ K suc h that h ′ ( H ′ ) is homogeneous in G and counta bly infinite. F rom this set, one obtains a regular ω -language H ⊆ L ′ ⊆ L with h ( H ) = h ′ ( H ′ ), i.e., h ( H ) is a homogeneous set of size ℵ 0 . This prov es (a). T o prov e (b), let ( L, h ) b e an injectiv e ω -automatic presentati on of s ome ( k , ℓ )-partition G = ( V , E 1 , . . . , E ℓ ) of size 2 ℵ 0 . Then there exists a regular ω -language L ′ ⊆ L with 542 DIETRICH KUSKE | L ′ | = ℵ 0 . Consider the sub-partition G ′ = ( h ( L ′ ) , E ′ 1 , . . . , E ′ ℓ ) with E ′ i = E i ∩ [ h ( L ′ )] k . T his ( k , ℓ )-partition has as ω -automatic p r esen tation the p air ( L ′ , h ). Then, by (a), there exists L ′′ ⊆ L ′ regular and infinite suc h that h ( L ′′ ) is homogeneous in G ′ and therefore in G . Since h is in ject iv e, this imp lies | h ( L ′ ) | = | L ′ | = ℵ 0 . Finally , w e sh o w (c) by a counterexa mple. Let L = { 0 , 1 } ω , V = L/ ∼ e , and h : L → V the canonical mapping. F urthermore, set E 1 = [ L ] k . Then G = ( V , E 1 , ∅ , . . . , ∅ ) is a ( k , ℓ )- partition with ω -automatic presentat ion ( L, h ). No w let H = S 1 ≤ i ≤ n U i V ω i ⊆ L for some non-empty languages U i , V i ⊆ { 0 , 1 } + suc h that h ( H ) is homogeneous and infinite. If | V ω i | = 1, then U i V ω i / ∼ e is finite. Sin ce h ( H ) is infinite, there exists 1 ≤ i ≤ n w ith | V ω i | > 1 implying the existence of words v , w ∈ V + i suc h that | v | = | w | and v 6 = w . F or u ∈ U i , the set u { v , w } ω ⊆ H has 2 ℵ 0 equiv alence classes wrt. ∼ e . Hence | h ( H ) | = 2 ℵ 0 . 3. Uncoun table homogeneous sets 3.1. A Ramsey theorem for ω -automatic (2 , ℓ ) -pa rt itions The main r esult of this section is the follo win g theorem that follo ws immed iately from Prop. 3.7 and L emm a 3.5. Theorem 3.1. F or al l ℓ ≥ 2 , we have (2 ℵ 0 , ω A ) → (2 ℵ 0 , co - ω erCF ∩ Λ ) 2 ℓ . 3.1.1. The pr o of. The pro of of this theorem will construct a language from co- ω erCF that describ es a homogeneous set. T his language is closely related to the follo win g language N = 1 { 0 , 1 } ω ∩ \ n ≥ 0 { 0 , 1 } n (0 { 0 , 1 } n 00 ∪ 10 n { 01 , 10 } ) { 0 , 1 } ω , i.e., an ω -w ord x b elongs to N iff it starts with 1 and , for ev ery n ≥ 0, we h a v e x [ n, 2 n + 3) ∈ 0 { 0 , 1 } ∗ 00 ∪ 10 ∗ 01 ∪ 10 ∗ 10. W e first list s ome useful pr operties of this language N : Lemma 3.2. The ω -language N is c ontaine d in (1 + 0 + ) ω , b elongs to co - ω erCF ∩ Λ , and supp( x ) ∩ supp( y ) is finite f or any x, y ∈ N distinct. Pr o of. Let b i ∈ { 0 , 1 } for all i ≥ 0 and su pp ose the wo rd x = b 0 b 1 . . . b elongs to N . Th en b 0 = 1, hence the wo rd x con tains at least one o ccurrence of 1. Note that, wh enev er b n = 1, then { b 2 n +1 , b 2 n +2 } = { 0 , 1 } , hence x conta ins infinitely man y o ccurrences of 1 and therefore infinitely man y o ccurrences of 0, i.e., N ⊆ (1 + 0 + ) ω . Note that the complement of N equals 0 { 0 , 1 } ω ∪ [ n ≥ 0  { 0 , 1 } n (0 { 0 , 1 } n { 01 , 10 , 11 } ∪ 1 { 0 , 1 } n { 00 , 11 } ) { 0 , 1 } ω  =   0 ∪ [ n ≥ 0 { 0 , 1 } n (0 { 0 , 1 } n { 01 , 10 , 11 } ∪ 1 { 0 , 1 } n { 00 , 11 } )   { 0 , 1 } ω . Since the expr essio n in squ are b rac k ets denotes a conte xt-free language, { 0 , 1 } ω \ N is an ev entually regular con text-free ω -language. IS RAMSEY’S THEOREM ω -AUTOMA TIC? 543 Note that a word 10 n 0 10 n 1 10 n 2 . . . b elongs to N iff, for all k ≥ 0, w e ha ve 0 ≤ n k − | 10 n 0 10 n 1 . . . 10 n k − 1 | ≤ 1. Hence, when b uilding a wo rd from N , we ha v e tw o c hoices for an y n k , say n 0 k and n 1 k with n 0 k < n 1 k . But then a 0 a 1 a 2 . . . 7→ 10 n a 0 0 10 n a 1 1 10 n a 2 2 . . . defines an order em b edd ing ( { 0 , 1 } ω , ≤ lex ) ֒ → ( N , ≤ lex ). Since ( R , ≤ ) ֒ → ( { 0 , 1 } ω , ≤ lex ), we get N ∈ Λ . No w let x, y ∈ N with sup p ( x ) ∩ sup p ( y ) infin ite . Th en th ere are arb itrarily long fi n ite w ords u and v of equal length suc h that u 1 and v 1 are p refixes of x and y , resp. Since u 1 is a prefix of x ∈ N , it is of th e f orm u 1 = u ′ 10 | u ′ | 1 (if | u | is ev en ) or u 1 = u ′ 10 | u ′ | 01 (if | u | is o dd) and analogously f or v . Ind u ctiv ely , one obtains u ′ = v ′ and therefore u = v . Since u and v are arbitrarily long, we sho w ed x = y . Lemma 3.3. L et ∼ and ≈ b e two e q u ivalenc e r elations on some set L such that any e quiva- lenc e c lass [ x ] ∼ of ∼ is c ountable and ≈ has 2 ℵ 0 e quivalenc e classes. Then ther e ar e elements ( x α ) α< 2 ℵ 0 of L such that [ x α ] ∼ e ∩ [ x β ] ≈ = ∅ f or al l α < β . Pr o of. W e construct the sequence ( x α ) α< 2 ℵ 0 b y ord inal ind u ctio n. So assum e we h a v e elemen ts ( x α ) α<κ for s ome ordinal κ < 2 ℵ 0 with [ x α ] ∼ ∩ [ x β ] ≈ = ∅ for all α < β < κ . Supp ose S α<κ [ x α ] ∼ ∩ [ x ] ≈ 6 = ∅ for all x ∈ L . F or x, y ∈ L with x 6≈ y , we h av e ( S α<κ [ x α ] ∼ ∩ [ x ] ≈ ) ∩ ( S α<κ [ x α ] ∼ ∩ [ y ] ≈ ) ⊆ [ x ] ≈ ∩ [ y ] ≈ = ∅ . Since S α<κ [ x α ] ∼ has κ · ℵ 0 ≤ max( κ, ℵ 0 ) < 2 ℵ 0 elemen ts, w e obtain | L | < 2 ℵ 0 , con tradicting | L | ≥ | L/ ≈ | = 2 ℵ 0 . Hence there exists an elemen t x κ ∈ L w ith [ x α ] ∼ ∩ [ x κ ] ≈ = ∅ f or all α < κ . Definition 3.4. Let u , v , and w b e nonempty words with | v | = | w | and v 6 = w . Define an ω -semigroup h omomo rphism h : { 0 , 1 } ∞ → Σ ∞ b y h (0) = v and h (1) = w and set H u,v ,w = u · h ( N ) where N is the set from Lemma 3.2 . Lemma 3.5. L et u , v , and w b e as in the pr e vious definition. Then H u,v ,w ∈ co - ω erCF ∩ Λ . Pr o of. Assume v < lex w . Then the mapping χ : { 0 , 1 } ω → Σ ω : x 7→ uh ( x ) (where h is the h omo morphism from the ab o ve definition) emb eds ( N , ≤ lex ) (and hence ( R , ≤ )) in to ( H u,v ,w , ≤ lex ). If w < lex v , then ( R , ≤ ) ∼ = ( R , ≥ ) ֒ → ( N , ≥ lex ) ֒ → ( H α,β ,γ , ≤ lex ). This pro v es that H u,v ,w b elongs to Λ . Since v 6 = w , the mapping χ is injectiv e. Hence Σ ω \ H α,β ,γ = Σ ω \ χ ( N ) = Σ ω \ χ ( { 0 , 1 } ω ) ∪ χ ( { 0 , 1 } ω \ N ) . Since χ can b e rea lized b y a generalize d sequen tial mac hine with B ¨ uc hi-acceptance, χ ( { 0 , 1 } ω ) is regular and χ ( { 0 , 1 } ω \ N ) (as the image of an ev en tu all y regular con text-free ω -language) is even tually regular con text-free. Hence Σ ω \ H u,v ,w is even tually regular con text-free. Prop osition 3.6. L e t G = ( L, E 0 , E 1 , . . . , E ℓ ) b e some (2 , 1 + ℓ ) -p artition with inje ctive ω -automatic pr esentation ( L, id) such that { ( x, y ) | { x, y } ∈ E 0 } ∪ { ( x, x ) | x ∈ L } is an e quivalenc e r e lation on L (denote d ≈ ) with 2 ℵ 0 e quivalenc e classes. Then ther e exist nonempty wor ds u , v , and w with v and w distinct, but of the same length, such that H u,v ,w is i -homo gene ous for some 1 ≤ i ≤ ℓ . Pr o of. There are finite ω -semigroups S and T and homomorphisms γ : Σ ∞ → S and δ : (Σ × Σ) ∞ → T suc h that (a) x ∈ L , y ∈ Σ ω , and γ ( x ) = γ ( y ) imply y ∈ L and (b) x, x ′ , y , y ′ ∈ L , { h ( x ) , h ( x ′ ) } ∈ E i , and δ ( x, x ′ ) = δ ( y , y ′ ) imply { h ( y ) , h ( y ′ ) } ∈ E i (for all 0 ≤ i ≤ ℓ ). 544 DIETRICH KUSKE By Lemma 3.3, there are words ( x α ) α< 2 ℵ 0 in L su c h that [ x α ] ∼ e ∩ [ x β ] ≈ = ∅ f or all α < β . In the f ol lo w ing, w e only need th e wo rds x 0 , x 1 , . . . , x C with C = | S | · | T | . Then [BKR08, Sections 3.1-3.3] 1 first constructs tw o ω -w ords y 1 and y 2 and an in finite sequen ce 1 ≤ g 1 < g 2 < . . . of natural num b ers su c h that in particular y 1 [ g 1 , g 2 ) < lex y 2 [ g 1 , g 2 ). Set u = y 2 [0 , g 1 ), v = y 1 [ g 1 , g 2 ), and w = y 2 [ g 1 , g 2 ). In the follo w ing, let h : { 0 , 1 } ∞ → Σ ∞ b e the h omomorphism from Def. 3.4 and s et χ ( x ) = uh ( x ) for x ∈ { 0 , 1 } ∗ . As in [BKR08], one can th en s h o w that all the words f rom H u,v ,w b elong to the ω -language L . In the follo w ing, set x ◦• = χ ((01) ω ) and x •◦ = χ ((10) ω ). Th en obvious alterations in th e pro ofs by B´ a r´ an y et al. sh o w: (1) [BKR08, Lemma 3.4] 2 If x, y ∈ { 0 , 1 } ω with supp( x ) \ supp( y ) and su pp ( y ) \ sup p ( x ) infinite, then { δ ( χ ( x ) , χ ( y )) , δ ( χ ( y ) , χ ( x )) } = { δ ( x •◦ , x ◦• ) , δ ( x ◦• , x •◦ ) } . (2) [BKR08, Lemma 3.5] x •◦ 6≈ x ◦• . There exists 0 ≤ i ≤ ℓ with { x •◦ , x ◦• } ∈ E i . Then (2) implies i > 0. Let x, y ∈ N b e distinct. Then s upp( x ) ∩ supp( y ) is fin ite b y Lemma 3.2. S in ce , on the other h and, supp( x ) and supp( y ) are b oth in fi nite, the tw o differences supp( x ) \ supp( y ) and sup p ( y ) \ sup p( x ) are infin ite. Hence we ob tain δ ( χ ( x ) , χ ( y )) ∈ { δ ( x •◦ , x ◦• ) , δ ( x ◦• , x •◦ ) } from (1). Hence (b) implies { χ ( x ) , χ ( y ) } ∈ E i , i.e., H u,v ,w is E i -homogeneous. Since H u,v ,w ∈ co- ω erC F ∩ Λ b y Lemma 3.5, the result follo ws. Prop osition 3.7. L et G = ( V , E ′ 1 , . . . , E ′ ℓ ) b e some (2 , ℓ ) -p artition with automatic pr esen- tation ( L, h ) . Then ther e exist u, v , w ∈ Σ + with v and w distinct of e qual length su ch that h ( H u,v ,w ) is homo gene ous and of size 2 ℵ 0 . Pr o of. T o apply Prop. 3.6, consider the follo win g (2 , 1 + ℓ )-partition G = ( L, E 0 , . . . , E ℓ ): • Th e underlying set is th e ω -language L , • E 0 comprises all sets { x, y } with h ( x ) = h ( y ) and x 6 = y , and • E i (for 1 ≤ i ≤ ℓ ) comprises all sets { x, y } with { h ( x ) , h ( y ) } ∈ E ′ i . Then ( L, id) is an injectiv e ω -automa tic presenta tion of th e (2 , 1 + ℓ )-partition G . By Prop. 3.6, there exists 1 ≤ i ≤ ℓ and wo rds u , v and w such that H u,v ,w is i -homogeneous in G . S ince ( E 0 , . . . , E ℓ ) is a partition of [ L ] 2 , w e hav e { x, y } / ∈ E 0 (and therefore h ( x ) 6 = h ( y )) for all x, y ∈ H u,v ,w distinct. Hence h is injectiv e on H u,v ,w . F ur thermore [ H u,v ,w ] 2 ⊆ E i implies [ h ( H u,v ,w )] 2 ⊆ E ′ i . Hence h ( H u,v ,w ) is an i -homogeneous set in G ′ of size 2 ℵ 0 . This finish es the p roof of Th eorem 3.1. 3.1.2. Effe ctive ness. Note that the pro of abov e is non-constructive a t sev eral p oin ts: Lemma 3.3 is n ot constru cti v e and the pro of prop er uses Ramsey’s theorem [BKR08, page 390] and mak es a Ramsey an factorisation coarser [BKR08, b egin of section 3.2]. W e n o w show that nev ertheless the wo rds u , v , and w can b e compu ted. By Prop. 3.7, it su ffices to decide f or a giv en triple ( u, v , w ) w hether h ( H u,v ,w ) is i -homogeneous for some fixed 1 ≤ i ≤ ℓ . T o b e more precise, let ( V , E 1 , . . . , E ℓ ) b e some (2 , ℓ )-partit ion with ω -automatic pr e- sen tation ( L, h ). F urthermore, let u, v , w ∈ Σ + with v 6 = w of the s ame length and w r ite H 1 The authors of [BKR08] requ ire [ x i ] ∼ e ∩ [ x j ] ≈ = ∅ for all 0 ≤ i, j ≤ C distinct, bu t they use it only for i < j . Hence we can apply their result here. 2 The auth ors of [BKR 08] only require one of the tw o differences to b e infinite, but th e pro of u ses that they b oth are infinite. IS RAMSEY’S THEOREM ω -AUTOMA TIC? 545 for H u,v ,w . W e hav e to decide whether H ⊆ L and H ⊗ H ⊆ L i ∪ L = . Note that H ⊆ L iff L ∩ Σ ω \ H = ∅ . But Σ ω \ H is con text-free, so the intersectio n is cont ext-free. Hence the emptiness of the in tersection can b e d eci ded. T o wards a decision of the second requirement , note that (Σ × Σ) ω \ ( H ⊗ H ) = (Σ ω \ H ⊗ Σ ω ) ∪ (Σ ω ∪ Σ ω \ H ) is the u nion of tw o conte xt-free ω -languages and therefore con text-free itself. Since L i ∪ L = is regular, the int ersection ( L i ∪ L = ) ∩ (Σ × Σ) ω \ ( H ⊗ H ) is con text-free imp lying that its emptiness is decidable. But this emptiness is equiv alen t to H ⊗ H ⊆ L 1 ∪ L = . 3.1.3. ω -automatic p artial or ders. ¿F rom Theorem 3.1, we no w der ive a necessary condition for a partial order of size 2 ℵ 0 to b e ω -automatic. A partial order ( V , ⊑ ) is ω -automatic iff there exists a regular ω -language L and a s u rjectio n h : L → V such that the relations R = = { ( x, y ) ∈ L 2 | h ( x ) = h ( y ) } and R ⊑ = { ( x, y ) ∈ L 2 | h ( x ) ⊑ h ( y ) } are ω -automatic. Corollary 3.8 ([BKR08] 3 ) . If ( V , ⊑ ) is an ω -automatic p artial or der with | V | ≥ ℵ 1 , then ( R , ≤ ) or an antichain of size 2 ℵ 0 emb e ds into ( V , ⊑ ) . Pr o of. Let ( V , ⊑ ) b e a partial ord er , L ⊆ Σ ω a regular ω -language and h : L → V a surjection suc h that R = and R ⊑ are ω -automatic. Define an injectiv e ω -automat ic (2 , 4)- partition G = ( L, E 0 , E 1 , E 2 , E 3 ): • E 0 comprises all p airs { x, y } ∈ [ L ] 2 with h ( x ) = h ( y ), • E 1 comprises all p airs { x, y } ∈ [ L ] 2 with h ( x ) ⊏ h ( y ) and x < lex y , • E 2 comprises all p airs { x, y } ∈ [ L ] 2 with h ( x ) ⊐ h ( y ) and x < lex y , and • E 3 = [ L ] 2 \ ( E 0 ∪ E 1 ∪ E 2 ) comprises all pairs { x, y } ∈ [ L ] 2 suc h that h ( x ) and h ( y ) are incomparable. F rom | L | ≥ | V | > ℵ 0 , w e obtain | L | = 2 ℵ 0 . Hence, by Prop. 3.6, there exists H ⊆ L 1-, 2- or 3-homogeneous with ( R , ≤ ) ֒ → ( H , ≤ lex ). S ince [ H ] 2 ⊆ E 1 ∪ E 2 ∪ E 3 and since G is a partition of L , th e mapping h acts injectiv ely on H . If [ H ] 2 ⊆ E 1 (the case [ H ] 2 ⊆ E 2 is symmetrical) then ( R , ≤ ) ֒ → ( H , ≤ lex ) ∼ = ( h ( H ) , ⊑ ). If [ H ] 2 ⊆ E 3 , then h ( H ) is an an tic h ain of size 2 ℵ 0 . A linear order ( L, ⊑ ) is sc atter e d if ( Q , ≤ ) cannot b e em b edded into ( L, ⊑ ). Automatic partial orders are defin ed similarly to ω -automatic partial orders with the help of fin ite automata instead of B ¨ uc hi-automata. Corollary 3.9 ([BKR08] 3 ) . A ny sc atter e d ω -automatic line ar or der ( V , ⊑ ) is c ountable. Henc e, • a sc atter e d line ar or der is ω -automatic if and only if it is automat ic, and • an or dinal α is ω -automatic if and only if α < ω ω . Pr o of. If ( V , ⊑ ) is not countable, then it embeds ( R , ≤ ) b y the pr evio us corollary and therefore in particular ( Q , ≤ ). The remaining tw o claims follo w immediately from [BKR08] (“coun table ω -automatic structures are automatic”) and [Del04] (“an ordin al is automatic iff it is prop erly smaller than ω ω ”), resp. 3 As p oin ted out by tw o referees, th e p ara graph b efore Sect. 4.1 in [BKR08] already hints at this result, although in a rather implicit w a y . 546 DIETRICH KUSKE Con trast T h eorem 3.1 with Th eorem 1.2: any u ncoun table ω -automatic ( k , ℓ )-partition con tains an uncounta ble homogeneous set of size 2 ℵ 0 . But w e we re able to pro v e this for k = 2, only . One would also wish the homogeneous set to b e r egular and not ju st from co- ω erCF. W e no w pro v e that these tw o shortcomings are un a v oidable: Theorem 3.1 do es not hold f or k = 3 nor is there alw ays an ω -regular h omog eneous set. These negativ e results hold ev en for injectiv e presentati ons. 3.2. A Sierpi ´ nski theorem for ω -automatic ( k , ℓ ) -partitions with k ≥ 3 W e first concentrate on the question whether some form of Theorem 3.1 holds f or k ≥ 3. The follo w ing lemma giv es the cen tral counte rexample for k = 3 and ℓ = 2, the b elo w th eorem then derives the general resu lt. Lemma 3.10. (2 ℵ 0 , ω iA ) 6→ ( ℵ 1 , ω LANG) 3 2 . Pr o of. Let Σ = { 0 , 1 } , V = L = { 0 , 1 } ω . F urther m ore, for H ⊆ L , we wr ite V H ∈ Σ ∞ for the longest common pr efix of all ω -w ords in H , V { x, y } is also written x ∧ y . Then let E 1 consist of all 3-sets { x, y , z } ∈ [ L ] 3 with x < lex y < lex z and x ∧ y < pref y ∧ z ; E 2 is th e complemen t of E 1 . This fi nishes the construction of the (3 , 2)-partition ( V , E 1 , E 2 ) of size 2 ℵ 0 with injectiv e ω -automatic present ation ( L, id). Note that 1 ∗ 0 ω is a count able E 1 -homogeneous set and that 0 ∗ 1 ω is a count able E 2 - homogeneous set. But th ere is no u n coun table homogeneous s et: First supp ose H ⊆ L is infinite and x ∧ y < pref y ∧ z f or all x < lex y < lex z fr om H . Let u ∈ Σ ∗ suc h that H ∩ u 0Σ ω and H ∩ u 1Σ ω are b oth nonempty and let x, y ∈ H ∩ u 0Σ ω with x ≤ lex y and z ∈ H ∩ u 1Σ ω . Then x ∧ y > pref u = y ∧ z and th er efore x = y (for otherwise, w e w ould ha v e x < lex y < lex z in H with x ∧ y > pref y ∧ z ). Hence we sh o w ed | H ∩ u 0Σ ω | = 1. Let u 0 = V H and H 1 = H ∩ u 0 1Σ ω . Since H ∩ u 0 0Σ ω is finite, the set H 1 is infin ite. W e pro ceed by in duction: u n = V H n and H n +1 = H n ∩ u n 1Σ ω satisfying | H n ∩ u n 0Σ ω | = 1. Then u 0 < pref u 0 1 ≤ pref u 1 < pref u 1 1 ≤ pref u 2 · · · with H = [ n ≥ 0 ( H ∩ u n 0Σ ω ) ∪ \ n ≥ 0 ( H ∩ u n 1Σ ω ) . Then any of the sets H ∩ u n 0Σ ω = H n ∩ u n 0Σ ω and T ( H ∩ u n 1Σ ω ) is a singleton, proving that H is coun table. Thus, there cannot b e an u ncoun table E 1 -homogeneous s et . So let H ⊆ L b e infinite with x ∧ y ≥ pref y ∧ z for all x < lex y < lex z . Since w e ha ve only t wo letters, w e get x ∧ y > pref y ∧ z for all x < lex y < lex z wh ic h allo ws to argue symmetrically to the ab o v e. T hus, ind eed, there is no u ncoun table homogeneous set in L . Theorem 3.11. F or al l k ≥ 3 , ℓ ≥ 2 , and λ > ℵ 0 , we have (2 ℵ 0 , ω iA ) 6→ ( λ, ω LANG ) k ℓ . Pr o of. Let G b e the (3 , 2)-partition from Lemma 3.10 that do es n ot hav e homogeneous sets of size λ and let ( L, id) b e an injectiv e ω -automatic presenta tion of G = ( V , E 1 , E 2 ) (in particular, V = L ). F or a set X ∈ [ L ] k , let X 1 < lex X 2 < lex X 3 b e th e three lexicographically least element s of X . Then set G ′ = ( V , E ′ 1 , E ′ 2 , . . . , E ′ ℓ ) with E ′ 1 = { X ∈ [ V ] k | { X 1 , X 2 , X 3 } ∈ E 1 } , E ′ 2 = { X ∈ [ V ] k | { X 1 , X 2 , X 3 } ∈ E 2 } , and E ′ i = ∅ f or 3 ≤ i ≤ ℓ . IS RAMSEY’S THEOREM ω -AUTOMA TIC? 547 Then ( L, id) is an injectiv e ω -automatic pr esen tation of G ′ . No w supp ose H ′ ⊆ L is homo- geneous in G ′ and of size λ . Then there exists H ⊆ H ′ of size λ suc h that f or any wo rds x 1 < lex x 2 < lex x 3 from H , there exists X ⊆ H ′ with X i = x i for 1 ≤ i ≤ 3 (if n ecessary , thro w a w a y some lexicographically largest elemen ts of H ′ ). Hence H is homogeneous in G , con trad icting Lemma 3.10. 3.3. C omplexit y of homogeneous sets in ω -automatic (2 , ℓ ) -partitions Ha ving sh o wn that k = 2 is a cen tral assumption in Th eo rem 3.1, w e n ow turn to the question whether homogeneous sets of lo w er complexit y can b e found. Construction. Let V = L denote th e regular ω -language (1 + 0 + ) ω . F urthermore, E 1 ⊆ [ L ] 2 comprises all 2-sets { x, y } ⊆ L such that supp( x ) ∩ sup p( y ) is fi nite or x ∼ e y . The set E 2 is the complement of E 1 in [ L ] 2 . This completes the construction of the (2 , 2)-partition G = ( L, E 1 , E 2 ). Note that ( L, id L ) is an inj ec tiv e ω -automatic pr esen tation of G . By Theorem 3.1, G has an E 1 - or an E 2 -homogeneous set of size 2 ℵ 0 . W e convince ourselv es that G has large homogeneous sets of b oth types. By Lemma 3.2, there is an ω - language N ⊆ (1 + 0 + ) ω of size 2 ℵ 0 suc h that the sup p orts of an y t wo w ord s from N hav e fin ite in tersection. Hence [ N ] 2 ⊆ E 1 and N has size 2 ℵ 0 . Bu t there is also an E 2 -homogeneous set L 2 of size 2 ℵ 0 : Note that the w ords from N are mutually non- ∼ e -equiv alen t and let L 2 denote the set of all w ords 1 a 1 1 a 2 1 a 3 . . . f or a 1 a 2 a 3 · · · ∈ N . Then for any x, y ∈ L 2 distinct, w e ha v e 2 N ⊆ supp( x ) ∩ supp( y ) and x 6∼ e y , i.e., { x, y } ∈ E 2 . Lemma 3.12. L et H ∈ LANG ∗ have si ze λ > ℵ 0 . Then H is not homo gene ous in G . Pr o of. By definition of LANG ∗ , there are languages U i , V i ∈ L ANG with H = S 1 ≤ i ≤ n U i V ω i . Since H is infi nite, there are 1 ≤ i ≤ n and x, y ∈ U i V ω i distinct with x ∼ e y and therefore { x, y } ∈ E 1 . Since | H | > ℵ 0 , th ere is 1 ≤ i ≤ n with | U i V ω i | > ℵ 0 ; w e set U = U i and V = V i . F rom | U | ≤ ℵ 0 , w e obtain | V ω | > ℵ 0 . Hence there are v 1 , v 2 ∈ V + distinct with | v 1 | = | v 2 | . Since uv ω 1 ∈ H and eac h element of H con tains infinitely man y o ccurrences of 1, the w ord v 1 b elongs to { 0 , 1 } ∗ 10 ∗ . Let u ∈ U b e arbitrary (su ch a w ord exists since U V ω 6 = ∅ ) and consider the ω -words x ′ = u ( v 1 v 2 ) ω and y ′ = u ( v 1 v 1 ) ω from U V ω ⊆ H . Th en x ′ 6∼ e y ′ since v 1 6 = v 2 and | v 1 | = | v 2 | . At the same time, su pp ( x ′ ) ∩ su pp( y ′ ) is in fi nite since v 1 con tains an o ccurrence of 1. Hence { x ′ , y ′ } ∈ E 2 . Th us, w e found ω -w ords x, y , x ′ , y ′ ∈ H with { x, y } ∈ E 1 and { x ′ , y ′ } / ∈ E 1 pro ving that H is not h omog eneous. Th us, w e fou n d a (2 , 2)-partition G = ( V , E 1 , E 2 ) with 2 ℵ 0 elemen ts and an injectiv e ω -automatic p r esen tation ( L, h ) such that (1) G h as sets L 1 and L 2 in co- ω erCF of size 2 ℵ 0 with [ L i ] 2 ⊆ E i for 1 ≤ i ≤ 2. (2) T h ere is no ω -language H ∈ LANG ∗ with H ⊆ L su c h that h ( H ) is homogeneous of size 2 ℵ 0 . Since all conte xt-free ω -languages b elong to LANG ∗ , the follo win g theorem follo ws the same w a y that Lemma 3.10 implied Theorem 3.11. Theorem 3.13. F or al l k , ℓ ≥ 2 and λ > ℵ 0 , we have (2 ℵ 0 , ω iA ) 6→ ( λ, ω CF ) k ℓ and (2 ℵ 0 , ω iA ) 6→ ( λ, ω REG) k ℓ . 548 DIETRICH KUSKE This r esult can b e un d erstoo d as another S ierp i´ n s ki theorem for ω -automatic ( k , ℓ )- partitions. T his time, it holds for all k ≥ 2 (not only for k ≥ 3 as Theorem 3.11). The pr ice to b e p ai d for this is the restriction of homogeneous sets to “simple” ones. In particular the n on-exi stence f regular h omog eneous sets pro vides a Sierpi ´ nski theorem in th e spirit of automatic structures. Op en questions Our p ositiv e result T h eorem 3.1 guarant ees the existence of some clique or anticl ique of size 2 ℵ 0 (and suc h a clique or antic lique can ev en b e constructed). But the follo win g situation is conceiv able: th e ω -automatic graph con tains large cliques without con taining large cliques that can b e d escribed by a language from co- ω erCF. In particular, it is not clear w hether the existence of a large clique is decidable. A related question concerns Ramsey quantifiers. Rub in [Rub08] has sho wn that the set of no des of an automatic graph whose neigh b ors conta in an infin ite an ticlique is regular (his result is m uc h more general, but th is formulatio n suffices for our purp ose). It is not clear whether this also h olds for ω -automatic graph s. A p ositiv e answer to this second question (assuming that it is effectiv e) would en tail an affirm at iv e answer to the decidabilit y question ab o ve . References [BG04] A. Blumensath and E. Gr¨ adel. Finite p rese ntations of infi nite structu res: Automata an d inter- pretations. The ory of Computing Systems , 37(6):641–674, 2004. [BKR08] V. B´ ar´ any , L. Kaiser, and S. Rubin. Cardinality and counting quantifiers on omega-automatic structures. In ST ACS’08 , pages 385–396 . IFIB Schloss Dagstuhl, 2008. [Blu99] A. Blumensath. Automatic structures. T echnical rep ort, R WTH A ac hen, 1999. [Del04] Ch. Delhomm´ e. Au tomatic it´ e des ordin aux et des graph es homog` enes. C. R. A c ad. Sci. Paris, Ser. I , 339:5–1 0, 2004. [ER56] P . Erd˝ os and R. R ado. A partition calculus in set theory . Bul l. AMS , 62:427– 489, 1956. [Gas98] W. Gasarc h . A survey of recu rsiv e combinatorics. 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