Hindman and Owings-like theorems without the Axiom of Choice

HINDMAN AND O WINGS-LIKE THEOREMS WITHOUT THE AXIOM OF CHOICE JOS ´ E ALBER TO GUZM ´ AN-VEGA, DA VID FERN ´ ANDEZ-BRET ´ ON, AND ELISEO SARMIENTO-ROSALES Abstract. W e inv estigate Hindman- and Owings-type Ramsey-theoretic state- ments in Zermelo–F raenk el set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dep endent Choice and/or the Axiom of Determinacy). W e study several v ariations of Hindman’s the- orem on Q -vector spaces; notably , we show that the uncountable analog of Hindman’s theorem fails for the additive group of R (under ZF ), and for Q - vector spaces of uncoun table dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-type conﬁgura- tions, we obtain several positive results, especially when assuming AD . These results highlight the in teraction betw een determinacy , algebraic structure, and dimension in the study of inﬁnite Ramsey theory without the Axiom of Choice. 1. Introduction Ramsey theory studies the emergence of structured mono chromatic conﬁgura- tions inside arbitrary colourings of large sets. Beginning with Ramsey’s theorem [19], a broad class of results can b e expressed via partition relations, and the modern viewp oin t emphasizes ho w set-theoretic strength and algebraic structure interact in inﬁnite combinatorics (see, e.g., [11, 12, 15]). In additiv e Ramsey theory , Hindman’s theorem [7] is a cen tral example: ev ery ﬁnite colouring of N admits an inﬁnite set X suc h that the set of all ﬁnite sums FS( X ) is mono chromatic. This phenomenon connects naturally with algebra in the Stone– ˇ Cec h compactiﬁcation and related structural metho ds [10]. A ma jor theme in the last decade has b een the exten t to which Hindman-type statemen ts p ersist b ey ond the countable setting. F ern´ andez-Bret´ on sho wed that Hindman’s theorem is essen tially coun table, in the sense that uncoun table analogues fail on an y comm uta- tiv e cancellativ e semigroup [2], and further strong failures for higher analogues w ere established in [4]. More recen tly , Hindman-like principles with uncountably man y colours and ﬁnite mono c hromatic conﬁgurations hav e b een in vestigated in [3, 13]. A second, closely related line of w ork concerns p airwise-sum (Owings-type) con- ﬁgurations. Owings [18] ask ed whether every ﬁnite colouring of N contains an inﬁnite set X such that X + X is mono chromatic. Despite some early partial progress [8], the problem remains op en to this day; versions of the same problem for R instead of N hav e been studied extensively: see for instance [9, 14, 16] and the developmen t of metho ds aimed at av oiding large-cardinal assumptions [20]. 2010 Mathematics Subje ct Classiﬁc ation. Primary 05D10, 03E60; Secondary 03E15, 20K01. Key wor ds and phrases. Axiom of Determinacy , Ramsey theory , Hindman’s theorem, Owings’ problem, partition relations, inﬁnite combinatorics, vector spaces ov er Q . 1 2 GUZM ´ AN-VEGA, FERN ´ ANDEZ-BRET ´ ON, AND SARMIENTO-ROSALES In this paper, w e revisit b oth Hindman- and Owings-type statemen ts in ZF , with- out assuming any choice-related principle when p ossible, but utilizing hypotheses suc h as the Axiom of Dep endent Choice DC or the Axiom of Determinacy AD when necessary . The original motiv ation for this w ork w as to study these Ramsey- theoretic principles under AD , or under AD + DC if necessary , but w e realized along the wa y that several of the pro ofs can actually b e carried out in ZF or in ZF + DC only . This will b e the criterion that informs the particular c hoice of additional assumptions b ey ond ZF . Hence, all of the pro ofs in this pap er take place in ZF , unless an extra assumption is expli citly noted in the header of the corresp onding theorem . Main con tributions. Our results fall into tw o families, reﬂecting the distinct nature of ﬁnite-sum and pairwise-sum conﬁgurations. Throughout the remainder of the pap er, we will refer to a “ κ - θ conﬁguration” to denote the relev ant Ramsey- t yp e statement for θ colours, where one attempts to obtain a mono c hromatic set generated by κ elements. • Hindman-lik e statements. W e b egin by showing p ositive results for b oth the ﬁnite-ﬁnite and inﬁnite-ﬁnite conﬁgurations on any Ab elian group; this largely follo ws directly from the original Hindman’s theorem, and only ZF is needed. In contrast, we obtain negativ e results for the uncountable-ﬁnite conﬁguration, b oth in R and in any Q -vector space with basis (the former in ZF , the latter in ZF + DC ); this highligts the role of cancellativit y and the rigid supp ort s tructure of such spaces (cf. [2]). The result for R in ZF is esp ecially revealing, b ecause it was formerly known only as a corollary of [2, Theorem 5], requiring the use of the Axiom of Choice. Along the w ay , w e also sho w another negative result for R regarding the ﬁnite-inﬁnite conﬁguration; while this result constitutes a particular case of [3, Theorem 12] in ZF C , it was not known b efore as a result in ZF only . Finally , w e obtain a negative result for the 3-inﬁnite conﬁguration on an y Q -v ector space with basis, generalizing [3, Theorem 5]; on the other hand, determinacy-driven partition prop erties at ω 1 and ω 2 yield a positive statement for the 2-inﬁnite conﬁguration, under AD , whenever we are dealing with a Q -vector space with a basis of cardinality ω 1 or ω 2 . • Owings-lik e statements. Under AD , we obtain a p ositiv e result for the inﬁnite-inﬁnite conﬁguration in R , placing Owings-type conﬁgurations in a determinacy setting and exploiting regularity phenomena av ailable under this axiom. In contrast, we obtain a negative result for the ﬁnite-inﬁnite conﬁguration in Q -v ector spaces with basis. Finally , regarding the inﬁnite- ﬁnite conﬁguration (the conﬁguration where most questions remain op en, ev en in the ZF C context), w e are able to obtain a p ositive result under AD for Q -vector spaces of dimension ω 1 or ω 2 , in the spirit of [16]. Organization of the pap er. Section 2 dev elops Hindman-lik e results: w e in- tro duce the ﬁnite-sums arrow notation, establish the simplest cases that follow directly from the classical Hindman’s theorem, and set up the structural lemmas used throughout; after that w e prov e the main negative and p ositive theorems for Q -v ector spaces. Section 3 treats Owings-like results: w e formalize the pairwise- sum arro w notation, pro v e the determinacy-based theorem for R under countable colourings, discuss limitations in cancellative groups, and conclude with the p ositiv e HINDMAN AND OWINGS-LIKE THEOREMS WITHOUT AC 3 theorem for Q -v ector spaces of dimension ω i ( i ∈ { 1 , 2 } ) for ﬁnitely many colours. There is a Section 4 where w e summarize the results obtained, the additional axioms needed for each, and prop ose further directions of study . 2. Hindman-like resul ts In this section w e analyze ﬁnite-sums partition relations in Ab elian groups, with particular emphasis on the additiv e group of R and other Q -v ector spaces. W e ﬁrst in tro duce the relev ant notation, brieﬂy discuss the simplest cases (the ﬁnite- ﬁnite and inﬁnite-ﬁnite conﬁgurations), and set up a combinatorial to ol that will allo w us to control supp orts of vectors. W e then prov e that uncoun table ﬁnite- sums homogeneity fails in R (in ZF ), and also in vector spaces of uncoun table dimension (with DC as an additional assumption). There is also a negative result for the 3-inﬁnite conﬁguration on Q -v ector spaces with basis. Finally , we contrast this negativ e phenomenon with p ositive ﬁnite conﬁgurations obtained from strong partition prop erties at ω 1 and ω 2 under AD . 2.1. Finite-sums notation and combinatorial to ols. As is customary in in- ﬁnite combinatorics, this text adopts Hungarian notation. Recall that, if X is a subset of some additiv e structure, FS( X ) denotes the set of ﬁnite sums from X , giv en by FS( X ) =  P a ∈ F a   F ⊆ X is ﬁnite and nonempty  . Deﬁnition 2.1. Let ( G, +) b e a commutativ e cancellativ e semigroup, and let κ, θ b e cardinals. W e write G → ( κ ) FS θ to denote the statement that for every colouring of G with θ colours, there exists a set M ⊆ G of cardinality κ such that FS( M ) is mono c hromatic. Hindman’s theorem [7] is the statement that N → ( ω ) FS θ for all ﬁnite θ . Another related statement (known to b e equiv alen t to Hindman’s theorem even b efore it w as prov ed) is that if [ ω ] <ω \ { ∅ } is ﬁnitely coloured, there is an inﬁnite pairwise disjoin t family B such that FU( B ) is mono c hromatic, where FU( B ) denotes the set of unions of ﬁnitely many elemen ts from B . Both of the statemen ts just mentioned can b e pro ven in ZF . F rom here, one can use a compactness argument to obtain ﬁnitary versions of the ab ov e statements: given ﬁnite θ and κ , there exists some natural num b er F such that, for every colouring of ℘ ( F ) \ { ∅ } in θ colours, there is a pairwise disjoin t family B ⊆ ℘ ( F ) \ { ∅ } of size κ such that FU( B ) is mono chromatic. Similarly (or b y mapping eac h ﬁnite a ⊆ N to the num b er P i ∈ a 2 i ), for eac h ﬁnite θ and κ , there is a num b er R (it suﬃces to take R = 2 F − 1 where F is as in the previous statemen t) suc h that, for every colouring of { 1 , · · · , R } in θ colours, there is M ⊆ R of size κ such that FS( M ) is mono chromatic. W e generalize the previous results to all Ab elian groups, th us settling the ﬁnite- ﬁnite and inﬁnite-ﬁnite conﬁgurations for Hindman’s theorem. R emark 2.2 . Theorems 2.3 and 2.4 can b e stated for all commutativ e cancellativ e semigroups, since ev ery comm utative cancellative semigroup embeds into a group. W e oﬃcially state them for Ab elian groups for the sak e of simplicity . 4 GUZM ´ AN-VEGA, FERN ´ ANDEZ-BRET ´ ON, AND SARMIENTO-ROSALES Theorem 2.3. F or al l ﬁnite κ, θ ther e exists an S ∈ N such that, for every A b elian gr oup G with | G | ≥ S , G → ( κ ) FS θ . Pr o of. Giv en ﬁnite κ and θ , take the corresp onding num b ers F , R as describ ed in the previous paragraph, and let S := max { R + 1 , R F + 1 } . Then, if G is an ab elian group with | G | ≥ S and c : G → θ , the following tw o cases are exhaustive: ∃ g ∈ G : o ( g ) > R : In this case, the set { g , 2 g , · · · , R g } is in bijection with { 1 , 2 , · · · , R } ; furthermore, this bijection preserves addition whenever the result of a sum stays within the set. Hence it suﬃces to let the restriction c ↾ { g , 2 g , . . . , Rg } induce a corresp onding colouring in { 1 , . . . , R } and apply the ﬁnite version of Hindman’s theorem. ∀ g ∈ G : o ( g ) ≤ R : Then we recursively choose h n ∈ G \ ⟨{ h i | i < n }⟩ ; the assumption implies that |⟨{ h i   i < n }⟩| ≤ R n and therefore there is alwa ys at least one such h n whenev er n ≤ F . By considering the map d : ℘ ( S ) \ { ∅ } → G given b y d ( f ) = P n ∈ f h n and applying the ﬁnite v ersion of the ﬁnite- unions theorem to c ◦ d : ℘ ( S ) → θ , there is a disjoint family B = { b i | i < κ } ⊆ ℘ ( F ) \ { ∅ } suc h that FU( B ) is c ◦ d -mono chromatic. This readily implies that, if we let M = { d ( b n ) | n < κ } , FS( M ) is c -mono chromatic. □ By essen tially the same argumen t, we are able to obtain the version of the the- orem where one obtains an inﬁnite mono chromatic set. Theorem 2.4. L et ( G, +) b e a De dekind-inﬁnite Ab elian gr oup. Then, for every ﬁnite θ , we have that G → ( ω ) FS θ . Pr o of. Giv en c : G → θ , we hav e tw o cases: ∃ g ∈ G : o ( g ) = ∞ : In this case, there is a semigroup isomorphism b et ween { ng   n ∈ N } and N , so it suﬃces to apply Hindman’s ﬁnite-sums theorem. ∀ g ∈ G : o ( g ) < ∞ : Since G is Dedekind inﬁnite, ﬁx an injection h : ω → G and recursively deﬁne i n := min( h [ ω ] \ ⟨{ h ( j ) | j < n }⟩ ); Then, the function d : [ ω ] <ω \ { ∅ } → G deﬁned by d ( f ) = P n ∈ f i n maps every ﬁnite union of disjoin t sets into the ﬁnite sum of the corresponding elements. Hence, by the ﬁnite-unions v ersion of Hindman’s theorem, there is a pairwise disjoint family B = { b n | n < ω } ⊆ [ ω ] <ω \ { ∅ } suc h that FU( B ) is d -monochromatic; setting M = { d ( b n ) | n < ω } , we must ha v e that FS( M ) is c -monochromatic. □ R emark 2.5 . As a corollary of Theorem 2.4, we obtain the following statement: assuming that ev ery inﬁnite set is Dedekind-inﬁnite, for every inﬁnite Ab elian group G and for ev ery ﬁnite θ w e hav e G → ( ω ) FS θ . In particular, the conclusion follows from each of the axioms of Countable Choice and Dep endent Choice. It is of note that the previous proof requires the assumption that G is Dedekind inﬁnite, as without it, the result ma y fail, ev en for inﬁnite Bo olean groups, as shown in [6, Theorem 5.2]. The remainder of this subsection is devoted to the study of ∆-systems and the ∆-system lemma, a combinatorial device that will b e crucial in what follows. Deﬁnition 2.6. A collection of sets D is a ∆ -system if there exists a set R (called the ro ot of the ∆-system) such that c ∩ d = R for any tw o distinct c, d ∈ D . HINDMAN AND OWINGS-LIKE THEOREMS WITHOUT AC 5 A stronger version of the following lemma is w ell-known in the ZF C context [1]; here we make explicit a w eaker v ersion that only requires DC . Lemma 2.7 ( DC ) . L et C b e an unc ountable c ol le ction of ﬁnite sets. Then ther e exists an inﬁnite ∆ -system D ⊆ C . Pr o of. Since DC implies (the Axiom of Countable Choice, which in turn implies) that coun table unions of countable sets are countable, w e deduce that there m ust be an n ∈ N such that uncountably man y elemen ts of C hav e cardinality n . So w e ma y assume without loss of generality that every element of C has the same cardinality n , and from here one can develop the usual pro of of the lemma b y induction on n ; w e sketc h the pro of for the b eneﬁt of the reader, and only explain the full details in the parts where the use of DC needs to b e made explicit. The base of the induction, i.e., the case n = 1, is immediate (in this case C is already a pairwise disjoin t family , that is, a ∆-system with empty ro ot). No w, if we assume n > 1, there are tw o cases, the easy one b eing if there exists an r suc h that r ∈ c for uncountably man y c ∈ C (in this case, apply the induction h yp othesis to obtain an inﬁnite ∆-system D ′ ⊆ { c \ { r }   r ∈ c ∈ C } , and simply let D = { d ∪ { r }   d ∈ D ′ } ). The more inv olv ed case is when, for ev ery r , there are at most coun tably man y c ∈ C with r ∈ C . This readily implies that, for every ﬁnite (or even countably inﬁnite) subset F ⊆ C , all but countably many elements of C are disjoint with ev ery elemen t of F (in case F is countably inﬁnite, one w ould use again DC to supp ort this claim). Hence, using DC , one can recursively ﬁnd an inﬁnite sequence of elements c m ∈ C with each c m disjoin t from all of the c k for k < m , so that D = { c m   m < ω } ⊆ C is an inﬁnite ∆-system (with empty ro ot). □ The imp ortance of Lemma 2.7 lies in its facilitation of several structural ar- gumen ts ov er Q -vector spaces. The atten tive reader will note that, in case C is assumed to b e a collection of subsets of some ordinal num b er, one do es not need the use of DC (since one is able to alwa ys choose minimums when necessary) and, in fact, the usual ∆-system lemma, where one obtains an uncountable ∆-system, holds in this context. Ho w ever, we will b e concerned primarily with the case where the elements of C are fully arbitrary . 2.2. Negativ e results. W e now sho w some failures of Hindman-type statemen ts, esp ecially regarding the uncoun table-ﬁnite conﬁguration. In the ZFC con text, these failures were prov ed in [2]; with some muc h stronger results established in [4]. Similar results for the inﬁnite-inﬁnite conﬁguration w ere established in [3, Theorem 12]. The ﬁrst surprising result is that, for the additiv e group of R , the same negativ e statemen ts can b e pro v ed in ZF . Theorem 2.8. R → (2) FS ω . Pr o of. Let c : R − → Z b e giv en by c ( r ) =      k if r ∈ [2 k , 2 k +1 ) , 0 if r = 0 , − k if r ∈ ( − 2 k +1 , − 2 k ] . Supp ose that r , s ∈ R are distinct and c ( r ) = c ( s ). W e will sho w that c ( r + s )  = c ( r ). One ma y assume without loss of generality that r, s > 0. Set c ( r ) = c ( s ) = k ∈ Z , then 2 k ≤ r , s < 2 k +1 , whic h implies 2 k +1 ≤ r + s < 2 k +2 . Hence c ( r + s ) = k + 1. □ 6 GUZM ´ AN-VEGA, FERN ´ ANDEZ-BRET ´ ON, AND SARMIENTO-ROSALES As a corollary of the previous result, we obtain the following negativ e result for uncoun table monochromatic sets o ver R ; a particular case of [2, Theorem 5] in ZFC , but somewhat surprising in the ZF context. Corollary 2.9. R → (uncountable) FS 2 . Pr o of. Consider the same colouring c as in the previous theorem and, given r ∈ R , let f ( r ) be the absolute v alue of c ( r ), that is, f ( r ) = k if | r | ∈ [2 k , 2 k +1 ) and f (0) = 0. W e now deﬁne the colouring d : R − → 2 b y letting d ( r ) ≡ f ( r ) mo d 2. Supp ose that X ⊆ R is an uncoun table set, whose elements ha v e the same colour under d . Since X is uncountable, the function c cannot b e one-to-one on X , so there are tw o distinct r , s ∈ X such that c ( r ) = c ( s ). No w, reasoning as in the previous theorem, we conclude f ( r + s ) = f ( r ) + 1, and so d ( r + s )  = d ( r ). □ W e now proceed to prov e a seemingly more general v ersion of the previous result; b ear in mind, how ev er, that we add the extra assumption of the existence of a basis (in ZF , one cannot guaran tee that R has a basis as a Q -vector space). The pro of is an adaptation of [2, Theorem 5] to the ZF context; the key to the argument lies in the algebraic structure of Q -vector spaces, as their cancellation prop erties prev ent the stabilization phenomena required for uncoun table ﬁnite-sums conﬁgurations. Theorem 2.10 ( DC ) . If V is a Q -ve ctor sp ac e with b asis, then V ↛ (uncoun table) FS 2 . Pr o of. Let B b e a basis of V , so that V ∼ = L b ∈B Q . Let us deﬁne supp : V → [ B ] <ω as follows: given v ∈ V , write v = P b ∈B q b b and let supp( v ) = { b ∈ B | q b  = 0 } . Then the colouring c : V − → 2 is deﬁned by letting c ( v ) ≡ ⌊ log 2 | supp( v ) |⌋ mo d 2. Pro ceeding by contradiction, suppose X ⊆ V is an uncoun table set suc h that FS( X ) is monochromatic, sa y in colour i . Since we are assuming DC , and therefore coun t- able unions of countable sets are coun table, there must b e an N ∈ N such that for uncountably many v ∈ X we ha ve | supp( v ) | = N , so assume without loss of generalit y that this holds of all v ∈ V . Similarly , letting for a v = P b ∈B q b b ∈ V , co ef ( v ) = { q b   b ∈ B } \ { 0 } , an analogous reasoning allo ws us to assume, without loss of generalit y , that there is a ﬁxed F ∈ [ Q \ { 0 } ] ≤ N suc h that ( ∀ v ∈ X )(co ef ( v ) = F ). Once again we use that countable unions of coun table sets are countable to notice that { supp( v )   v ∈ X } is an uncountable set, so we ma y use Lemma 2.7 together with DC to get a sequence { v n | n < ω } ⊆ X with pairwise distinct supp orts such that { supp( v n )   n < ω } forms a ∆-system, sa y with root R . Since there are at most | R | | F | (a ﬁnite num b er) coeﬃcients for the v n in the p ositions of the ro ot R , b y the pigeonhole principle, we may assume that the co eﬃcients of the v n in R are constan t, guaranteeing that when adding vectors of W the terms corresp ond- ing to R do not cancel. This implies that, if v n 1 , · · · , v n L ∈ X are distinct, then    supp  P L i =1 v n i     = | R | + L ( N − | R | ). So the cardinalities of the supports of elemen ts of FS( X ) form an inﬁnite arithmetic progression (with base N and diﬀer- ence N − | R | ); letting m b e such that 2 m ≤ N < 2 m +1 and n ≤ m b e such that 2 n ≤ N − | R | < 2 n +1 , we see that 2 m +1 = 2 m − n 2 n + 2 m ≤ 2 m − n ( N − | R | ) + N ≤ 2 m − n 2 n +1 + 2 m +1 = 2 m +2 , so that any ﬁnite sum of 2 m − n + 1 elements from V must ha v e a supp ort with cardinalit y | R | + (2 m − n + 1)( N − | R | ) = 2 m − n ( N − | R | ) + N . So if v ∈ X is arbitrary , and w ∈ FS( X ) is any s um with 2 m − n + 1 summands, w e must hav e c ( v ) ≡ m mo d 2 and c ( w ) ≡ m + 1 mo d 2, contradicting that FS( X ) is mono chromatic. □ HINDMAN AND OWINGS-LIKE THEOREMS WITHOUT AC 7 The preceding theorems sho w that the ﬁnite-sums phenomenon remains essen- tially countable in many Q -vector spaces. Mo ving on to the ﬁnite-inﬁnite c onﬁguration, w e record an additional ﬁnite fail- ure which further illustrates the rigidity imp osed b y cancellation. A version of the follo wing result was established in [3, Theorem 5] in the ZFC context; the pro of b elo w do es not use AC . Theorem 2.11. L et V b e a Q -ve ctor sp ac e with b a sis. Then V → (3) FS ω . Pr o of. Giv en a basis B for V , we deﬁne the function ⟨ | ⟩ : V × V − → Q by ⟨ v | w ⟩ = P b ∈B x b y b , where v = P b ∈B x b b and w = P b ∈B y b b ; all of these sums are w ell-deﬁned b ecause x b = y b = 0 for all but ﬁnitely man y b ∈ B . It is hard not to see that the function just deﬁned satisﬁes all the prop erties of an inner product; we no w pro ceed to deﬁne the colouring c : V − → Q by c ( v ) = ⟨ v | v ⟩ . Supp ose that there are three distinct vectors u, v , w ∈ V such that FS( { u, v , w } ) is mono chromatic, say in colour q ∈ Q . Then w e must hav e q  = 0 and, in addition, q = ⟨ u | u ⟩ = ⟨ u + v | u + v ⟩ = ⟨ u | u ⟩ + ⟨ v | v ⟩ + 2 ⟨ u | v ⟩ , from where one can deduce that ⟨ u | v ⟩ = − 1 2 q . In an entirely similar manner one sho ws that ⟨ v | w ⟩ = ⟨ u | w ⟩ = − 1 2 q , and therefore ⟨ u + v + w | u + v + w ⟩ = ⟨ u | u ⟩ + ⟨ v | v ⟩ + ⟨ w | w ⟩ + 2  ⟨ u | v ⟩ + ⟨ v | w ⟩ + ⟨ u | w ⟩  = 3 q + 2  − 3 2 q  = 0 , meaning in particular that c ( u + v + w ) = 0  = q = c ( u ), a contradiction. □ 2.3. P ositiv e ﬁnite conﬁgurations from partition prop erties. Despite the negativ e results ab ov e, strong partition prop erties av ailable under determinacy yield p ositiv e ﬁnite-sums conﬁgurations in suﬃcien tly large dimensions; although for this w e do need to assume AD . The pro of b elow, repro duced here for the b eneﬁt of the reader, is a classic argumen t made many times ov er to deduce some small mono c hromatic ﬁnite-sums c onﬁgurations from Ramsey-type prop erties, using the fact that AD ensures that ω 1 , ω 2 ha ve large-cardinal prop erties. Deﬁnition 2.12. Let λ , κ, ι and θ b e cardinals. W e write λ → ( κ ) ι θ for the state- men t: “for every colouring of [ λ ] ι in to θ colours, there is a mono chromatic subset of size κ ,” i.e. ( ∀ c : [ λ ] ι → θ ) ( ∃ M ∈ [ λ ] κ )  | c [[ M ] ι ] | = 1  . Theorem 2.13. Every me asur able c ar dinal κ is we akly c omp act [11, Lemma 10.18] and thus satisﬁes κ → ( κ ) ι θ for every ι < ω and every θ < κ [12, Theorem 7.8] . In order to reinforce the idea that AD pro vides strong Ramsey-type consequences, w e recall the follo wing classical results. Theorem 2.14 (Martin [12, Theorem 28.2]) . Under ZF + AD , ω 1 is me asur able. Mor e over, ω 1 → ( ω 1 ) ω 1 2 ; henc e ω 1 → ( ω 1 ) ω 1 θ for every θ < ω 1 . Theorem 2.15 (Solov a y [12, Theorem 28.6]) . Under ZF + AD , ω 2 is me asur able. 8 GUZM ´ AN-VEGA, FERN ´ ANDEZ-BRET ´ ON, AND SARMIENTO-ROSALES Theorem 2.16 ( AD ) . L et V b e a Q -ve ctor sp ac e of wel l-or der able dimension. (1) If dim( V ) = ω 1 , then V → (2) FS ω . (2) If dim( V ) = ω 2 , then V → (2) FS ω 1 . Pr o of. Let B = { b i | i < κ } (where κ is either ω 1 or ω 2 , accordingly) b e a basis of V and c : V → λ (where λ is either ω or ω 1 , accordingly). W e deﬁne d : [ κ ] 2 → λ b y d ( { α, β } ) = c ( b β − b α ) whenever α < β . Then, since κ is measurable under AD and therefore κ − → ( κ ) 2 λ , there exists an M ⊆ κ with | M | = κ such that [ M ] 2 is d - mono c hromatic, say in colour i . Let α, β , γ ∈ M ; assume without loss of generality that α < β < γ , and deﬁne v = b β − b α and w = b γ − b β . Then, w e hav e c ( v ) = c ( b β − b α ) = d ( { α, β } ) = i c ( w ) = c ( b γ − b β ) = d ( { β , γ } ) = i c ( v + w ) = c ( b γ − b α ) = d ( { α, γ } ) = i, that is, FS( { v , w } ) is c -mono chromatic (in colour i ). □ R emark 2.17 . Recall that, by Martin’s theorem, ω 1 → ( ω 1 ) 2 2 ω . Therefore, by using the same argument as in the previous pro of, we may conclude (under AD ) that, for V a Q -vector space of dimension ω 1 , we hav e V → (2) FS 2 ω . 3. O wings-like resul ts W e now turn to pairwise-sum conﬁgurations, in the spirit of Owings’ problem. In contrast with the ﬁnite-sums phenomena studied in the previous section, the b eha viour of M + M under colourings exhibits a diﬀerent interaction b etw een descriptiv e-set-theoretic regularity and algebraic structure. W e b egin b y introducing the relev ant notation and recalling the known bac k- ground in ZFC . W e then establish a p ositive result for R under dep endent choice, deriv ed from regularity prop erties. Finally , we analyze algebraic limitations in can- cellativ e groups and present p ositive results for large-dimensional Q -v ector spaces obtained from strong partition prop erties. 3.1. P airwise-sum notation and background. Deﬁnition 3.1. Let ( G, +) be a commutativ e semigroup, and let κ, θ b e cardinals. W e write G → ( κ ) · + · θ to denote the statement that for every colouring of G in θ colours, there exists M ⊆ G , | M | = κ , such that M + M is mono chromatic. In the previous deﬁnition, M + M denotes the set { a + b   a, b ∈ M } . Owings’s problem [18] asks whether N → ( ω ) · + · 2 ; this problem has b een op en since 1974 and, surprisingly , the main diﬃculty seems to lie on the requirement of using precisely t wo colours, since Hindman [8] has prov ed that N ↛ ( ω ) · + · 3 . Recently , there has b een an imp ortan t amount of work devoted to studying these types of Owings- t yp e problems in the setting of the additive group R , and now ada ys it is kno wn that R → ( ω ) · + · 2 holds [20], and that the statement ( ∀ k < ω )( R → ( ω ) · + · k ) is both indep enden t from [9], and consistent with [14, 20], the ZFC axioms. So we pro ceed to study these kinds of statemen ts, under ZF plus p ossibly other assumptions, for some particular cases of G (mostly either R , or Q -v ector spaces with basis), for some sp eciﬁc conﬁgurations of κ , θ . HINDMAN AND OWINGS-LIKE THEOREMS WITHOUT AC 9 It is w orth noting that results ab out the ﬁnite-ﬁnite conﬁguration are already kno wn completely , since [5, Theorem 1] completely characterizes those G for which it is the case that ( ∀ n, r < ω )( G → ( n ) · + · r ). The pro of of this result, belonging completely to the ﬁnite realm, do es not use any version of the Axiom of Choice. In what follows, we explore other conﬁgurations. 3.2. P ositiv e results on R from regularit y. Hindman, Leader and Strauss [9] pro ved, in the ZFC con text, that if N ⊆ R is a Baire-measurable nonmeagre set, then there is an X ∈ [ R ] ω 1 suc h that X + X is mono c hromatic. Their proof only uses the Axiom of Choice when recursively c ho osing the “next” element of X , but the argumen t that suc h next elemen t exists can be carried out in ZF only . Therefore we ma y assume a w eaker h ypothesis, such as DC , and use an entirely similar argument to obtain a countable suc h set X . W e only sketc h the pro of for completeness, and refer the reader interested in some more sp eciﬁc details to [9, Lemma 4.1 and Theorem 4.2]. Lemma 3.2 ( DC ) . If N ⊆ R is nonme agr e and Bair e-me asur able, then ther e exists an X ∈ [ R ] ω such that X + X ⊆ N . Pr o of. Since N is Baire-meas urable, we may write N = A △ M , with A open and M meagre. Without loss of generality , by replacing N with a translate if necessary , w e may assume that 0 ∈ A (since, if we are able to ﬁnd X + X ⊆ c + N , letting Y = − c 2 + X it is readily c hec ked that Y + Y ⊆ N ). Pic k a δ > 0 such that (0 , 2 δ ) ⊆ A and c ho ose some x 0 ∈ (0 , δ ) \ 1 2 M , it immediately follo ws that { x 0 } + { x 0 } = { 2 x 0 } ⊆ (0 , 2 δ ) \ M ⊆ N . Using DC , we recursiv ely choose the remaining sequence of x n as follows: assuming by induction h yp othesis that X n = { x k   k < n } is such that X n + X n ⊆ N , we ha v e (0 , δ ) ⊆ 1 2 A ∩  \ k

Original Paper

Loading high-quality paper...

Comments & Academic Discussion

Loading comments...

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment