Graham conjecture on small sets in abelian groups

GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GR OUPS SIMONE COST A, STEF ANO DELLA FIORE, MA TTIA FONT ANA, AND LLU ´ IS VENA Abstract. A famous conjecture of Graham asserts that ev ery set A ⊆ Z p \ { 0 } can be ordered so that all partial sums are distinct. Although this conjecture was recently pro ved for sufficien tly large primes b y Pham and Sauermann in [17] (combined with earlier results of [6]), it remains op en for general ab elian groups, ev en in the cyclic case Z k . In this pap er, using a recursiv e approach, w e inv estigate the sequenceabilit y of subsets A in generic ab elian groups for small v alues of | A | . W e pro ve that any subset A ⊆ G \ { 0 } with | A | ≤ 20 is sequenceable where previously it was known only for | A | ≤ 9. This b ound is impro ved to | A | ≤ 22 for zero-sum subsets. Finally , regarding the related CMPP conjecture, w e show that zero-sum subsets without inv erse pairs are sequenceable for | A | ≤ 23. 1. Introduction Let A b e a finite subset of an ab elian group ( G, +). W e sa y that an ordering a 1 , . . . , a | A | of A is valid if its partial sums p 1 = a 1 , p 2 = a 1 + a 2 , . . . , p | A | = a 1 + · · · + a | A | are pairwise distinct. Moreo v er, this ordering is se quencing if it is v alid and p i  = 0 for every 1 ≤ i ≤ | A | − 1. A set is said to b e se quenc e able if it admits a sequencing ordering. In the literature, there are several conjectures ab out v alid and sequencing orderings. W e refer to [10, 15, 16] for an ov erview of the topic, [2 – 4, 9] for lists of related conjectures, and [7] for a treatment using rainbow paths. Here, w e first recall the conjecture Graham p osed for groups Z p (and more generally , posed for ab elian groups b y Alspach). Conjecture 1.1 (Graham/Alspac h conjecture, see [13], [12] and [3]) . L et G b e an ab elian gr oup. Then every subset A ⊆ G \ { 0 } is se quenc e able. The follo wing related conjecture w as p osed b y Costa, Morini, P asotti and P ellegrini k eeping in mind applications to Heffter arrays. Conjecture 1.2 (CMPP conjecture, see [10]) . L et G b e an ab elian gr oup. Then every subset A ⊆ G \ { 0 } whose elements sum to zer o and such that { x, − x } ⊂ A for any x ∈ G , is se quenc e able. Un til recently , the main results on these conjectures were for small v alues of | A | ; in par- ticular, in [9], the Graham’s conjecture was pro v ed, using the Combinatorial Nullstellensatz theorem, for sets A ⊆ Z p of size at most 12. F or generic abelian groups G the b est result is due to Alspac h and Liversidge ([3]) who prov e, via a brute-force p oset-approach, the conjecture for sets A ⊆ G \ { 0 } of size at most 9. Regarding the CMPP conjecture, it w as prov ed for sets A ⊆ Z p of size at most 13 and for sets A ⊆ G \ { 0 } of size at most 10. The first result inv olving arbitrarily large sets A w as presented by Kra vitz [14], who used a rectification argument to show that Graham’s conjecture holds for all sets A of size | A | ≤ log p/ log log p . Sa win [18] also prop osed a similar argument (but was not published). This 2020 Mathematics Subje ct Classific ation. 11B75. Key words and phr ases. Sequenceability . 1 2 GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS b ound w as then impro v ed in [5] b y Bedert and Kra vitz, and, very recen tly , by Costa and Della Fiore in [8], who pro ved, again using a rectification based approac h that: Theorem 1.3 ([8]) . L et p b e the le ast prime divisor of k . Then ther e exists a c onstant c > 0 such that every subset A ⊆ Z k \ { 0 } is se quenc e able pr ovide d that | A | ≤ exp  c (log p ) 1 / 3  . As mentioned ab ov e, the most impressiv e result on sequenceability problems was pro v ed in [17], where, com bined with earlier results of [6], they settled Graham’s conjecture on Z p for all sufficien tly large primes p . This result is a consequence of anticoncen tration inequalities dev elop ed using a discrete F ourier approach that seems hard to adapt to the cyclic case. Moreo v er, the condition that p is sufficien tly large leav es op en all small primes. In this pap er, we inv estigate Conjectures 1.1 and 1.2 for small v alues of | A | and in generic ab elian groups G . In particular, using a recursiv e approach, w e ha v e b een able to prov e the follo wing. Theorem 1.4. L et G b e an ab elian gr oup. Then every subset A ⊆ G \ { 0 } is se quenc e able pr ovide d that | A | ≤ 20 . If we assume, moreo ver, that P x ∈ A x = 0, we can impro v e this bound. Theorem 1.5. L et G b e an ab elian gr oup and let A ⊆ G \ { 0 } b e such that P x ∈ A x = 0 . Then A is se quenc e able pr ovide d that | A | ≤ 22 . Finally , in case of the CMPP conjecture, we can go one size further. Theorem 1.6. L et G b e an ab elian gr oup and let A ⊆ G \ { 0 } b e such that P x ∈ A x = 0 and { x, − x } ⊂ A for any x ∈ G . Then A is se quenc e able pr ovide d that | A | ≤ 23 . The results obtained here significantly improv e the previously known b ounds for general ab elian groups. Indeed, while the conjecture was previously verifi ed only for sets of size at most 9 in arbitrary abelian groups, our recursive approac h allo ws us to extend this bound up to 20, and ev en further in the presence of additional structural assumptions. The k ey ingredien t of our metho d is a Recursive Lemm a, presen ted in Section 2 as Corol- lary 2.5, allo wing the reduction of the problem to smaller sets via a suitable merging op eration. In this section, we will first present a polynomial pro of of this lemma for the groups Z p and then a more general direct pro of. The general pro of relies on the characterization of the sub- sets of G closed under the sum of distinct elements, whic h is also giv en in Section 2. Finally , in the last Section, we will discuss and describ e the computational approach used to get our main theorems. 2. The Recursive Lemma In this section, we prov e a Recursive Lemma that is crucial in our recursiv e approach for the case of a sum-zero subset A ⊂ Z k . Namely , we prov e that we can choose x 1 , x 2 ∈ A so that x 1 + x 2 is nonzero and do es not b elong to A \ { x 1 , x 2 } . This allo ws us to w ork recursiv ely with A ′ := ( A \ { x 1 , x 2 } ) ∪ { x 1 + x 2 } . GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS 3 2.1. A Com binatorial Nullstellensatz’s approac h in Z p . W e presen t a p olynomial-based pro of of the Recursiv e Lemma. Although w e will later prov e this lemma in a more general con text, we b eliev e this particular approac h is elegant and mathematically significant. The pro of relies on the following v ersion of Alon’s Com binatorial Nullstellensatz: Theorem 2.1. [1, The or em 1.2] L et F b e a field and let f = f ( x 1 , . . . , x k ) b e a p olynomial in F [ x 1 , . . . , x k ] . Supp ose the de gr e e of f is k P i =1 t i , wher e e ach t i is a nonne gative inte ger, and supp ose the c o efficient of k Q i =1 x t i i in f is nonzer o. Then, if A 1 , . . . , A k ar e subsets of F with | A i | > t i , ther e ar e a 1 ∈ A 1 , . . . , a k ∈ A k so that f ( a 1 , . . . , a k )  = 0 . In order to apply this theorem for pro ving the existence of the elements x 1 , x 2 that can b e merged, we construct a suitable homogeneous p olynomial F k of degree ( k +2)( k − 1) 2 , identifying a monomial with nonzero co efficient suc h that the degree of eac h of its terms x i is less than | A | = k . Here we will assume that p > 25 and 23 ≥ k ≥ 10 as the case p ≤ 25 has already b een solv ed in [10], the case k ≤ 9 has been solved in [3], and our recursiv e approac h can reac h k up to 23. W e define: F k ( x 1 , . . . , x k ) = ( x 1 + x 2 ) k Y i =3 ( x 1 + x 2 − x i ) Y i,j ∈ [1 ,k ] ,i>j ( x i − x j ) . The existence of v alues x 1 , . . . , x k ∈ A suc h that F k ( x 1 , . . . , x k )  = 0, giv en b y Theorem 2.1, implies that x 1 + x 2 is nonzero and it do es not belong to A \ { x 1 , x 2 } . W e can use Alon’s com binatorial Nullstellensatz to pro v e this statemen t. Indeed, assuming p > 25 we find, with Mathematica, for an y k ∈ { 10 , . . . , 23 } a monomial of the form Q k i =1 x t i i in the dev elopment of F k whose coefficient is nonzero and suc h that k > t i for all i ∈ { 1 , . . . , k } (see T able 1 for the explicit co efficients). W e obtain then the Recursiv e Lemma in the Z p case. Lemma 2.2. L et A ⊆ Z p \ { 0 } b e of c ar dinality 10 ≤ k ≤ 23 . Then ther e exist two distinct elements x 1 , x 2 ∈ A so that A ′ := ( A \ { x 1 , x 2 } ) ∪ { x 1 + x 2 } is a subset of Z p \ { 0 } of size k − 1 . k Co efficien t in Z 10 2 2 · 11 11 2 · 3 3 12 − 5 · 13 13 − 7 · 11 14 2 · 3 2 · 5 15 2 3 · 13 16 − 7 · 17 17 − 3 3 · 5 18 2 3 · 19 19 2 · 5 · 17 20 − 3 3 · 7 21 − 11 · 19 22 2 · 5 · 23 23 2 2 · 3 2 · 7 T able 1. Coefficient of the monomial x k − 1 1 x 2 x 2 3 · · · x k − 1 k in F k for 9 ≤ k ≤ 23. 4 GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS 2.2. A direct pro of. In this subsection we pro v e the Recursiv e Lemma in a muc h more general setting. More precisely w e characterize the sets A for whic h a i + a j b elongs to A ∪ { 0 } whenev er i and j are distinct indices. Prop osition 2.3 sho ws that, for most cases, one of the elemen ts can be c hosen freely . The c haracterization, Theorem 2.4, and the next prop osition, Prop osition 2.3, follo ws from the same ideas leading to Kneser’s theorem [11, 19, 20], which claims that if G is an ab elian group, then | A + B | ≥ | A | + | B | − 1 unless A + B is a union of cosets of a subgroup. Prop osition 2.3 (T ranslation by a i ) . L et A = { a 1 , . . . , a k } ⊂ Z m \ { 0 } b e a set of p airwise differ ent elements. Then, for e ach i ∈ [ k ] , one of the fol lowing holds. (1) Ther e exists a j  = i such that a i + a j / ∈ { 0 , a 1 , . . . , a k } . (2) Ther e exists an s < k such that A ∪ { 0 } = { a i , 0 , − a i , − 2 a i , . . . , − sa i } ∪ M , wher e M is a union of c osets of the sub gr oup gener ate d by a i . If M is the empty set, s = k − 1 . Pr o of of Pr op osition 2.3. Let g i : Z m → Z m b e the mapping g i ( x ) = x + a i . Then | g i ( { a 1 , . . . , a k } ) | = k . F urther, since 0 / ∈ { a 1 , . . . , a k } , then a i / ∈ g i ( { a 1 , . . . , a k } ). Let b 0 , . . . , b s b e the longest chain of distinct elements of { a 1 , . . . , a k } so that g i ( b j ) = b j +1 for eac h j ∈ { 0 , . . . , s − 1 } . If g i ( b s ) ∈ { a 1 , . . . , a k } , then g i ( b s ) = b 0 as the map g i is injectiv e, and we obtain g i ( b s ) = b s + a i = b 0 and since b s = g s i ( b 0 ) = sa i + b 0 , then ( s + 1) a i + b 0 = b 0 whic h implies that ( s + 1) a i = 0, and since s + 1 is the minimal with such prop ert y , then s + 1 is the order of a i . F urther, if w e let S = ⟨ a i ⟩ b e the subgroup generated b y a i , then { b 0 , . . . , b s } + S = { b 0 , . . . , b s } and it is a coset of S . Note that { b 0 , . . . , b s } ⊂ { a 1 , . . . , a k } and th us it is not the 0 coset (or the subgroup S ), as it do es not con tain the identit y elemen t 0. W e consider the new set { a 1 , . . . , a k } \ { b 0 , . . . , b s } and apply the result inductively . Observ e that this pro cedure of iteratively applying g i cannot reach a i , since the mapping is bijective, and that w ould mean that w e apply g i to 0, but 0 is not in A , and thus we nev er consider it as an element in the domain of the mapping. In particular, s ≤ k − 2. If g i ( b s ) = 0, and s = k − 2. Then we conclude that b 0 + a i = b 1 , b 1 + a i = b 2 and in general b j = b 0 + j a i → b 0 = b j − j a i with b s = b 0 + ( k − 2) a i = 0 → b s − ( k − 2) a i = b 0 , and b s + a i − ( k − 1) a i = b 0 = − ( k − 1) a i . Then A = { a 1 , . . . , a k } = { b 0 , . . . , b s , a i } as s = k − 2, and { b 0 , . . . , b s , a i } = {− ( k − 1) a i , . . . , − a i , a i } and so the second part of the statemen t follows with M as the empt y set. If g i ( b s ) = 0, and s < k − 2, this means that A \ { a i , b 0 , . . . , b s } is non-empt y . Observe also that g i ( A \ { b 0 , . . . , b s } ) ∩ { b 0 , . . . , b s } = ∅ since the mapping g i is bijectiv e and all the elemen ts in { b 1 , . . . , b s } ha ve a preimage already , and g i ( x )  = b 0 for eac h x ∈ A \ { a i , b 0 , . . . , b s } b y the assumed maximality on the s of { b 0 , . . . , b s } . By applying the same pro cedure of finding a (new) maximal c hain { c 0 , . . . , c r } of the elemen ts of A \ { a i , b 0 , . . . , b s } ; we conclude that we are either in the first case ( g i ( c r ) = c 0 ) and find another coset of the subgroup, and w e w ould again apply the result recursively , or c r + a i is a new elemen t differen t from 0 and the first part of the statement follo ws. (Note that g i ( b s ) = 0 already , and th us the other tw o cases do not apply). □ The Recursive Lemma is then a consequence of the follo wing result. Theorem 2.4 (Characterization Theorem) . L et G b e an ab elian gr oup and let A = { a 1 , . . . , a k } ⊆ G \ { 0 } b e a set of k ≥ 2 elements such that, for any distinct i, j , a i + a j ∈ { 0 , a 1 , . . . , a k } . Then A ∪ { 0 } is a sub gr oup of G . Pr o of of The or em 2.4. First of all, we consider the case where A is constituted only by in v olu- tions. In this case, it is clear that A ∪ { 0 } is inv erse-closed and it is closed under the sum of its GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS 5 elemen ts and is hence a subgroup of G . Similarly , also when | A | = 2 w e hav e that A ∪ { 0 } is in v erse-closed and it is closed under the sum. So in the following we will assume the existence of a non-inv olution elemen t and that | A | ≥ 3. No w w e pro v e that, under these hypotheses, there exists a non-inv olution a i ∈ A so that − a i ∈ A . Indeed, assume otherwise. With the h yp othesis for any distinct i, j , a i + a j ∈ { 0 , a 1 , . . . , a k } w e conclude that for an y distinct { i, j } , a i + a j  = 0. There are  k 2  suc h pairs and only n elemen ts in A , so eac h element of A must b e the sum of an av erage of  k 2  k = ( k − 1) / 2 distinct pairs. If a sum would hav e more than this num b er of pairs, then t w o such pairs w ould ha v e an element in common, thus making the t w o pairs the same. So eac h element of A must b e the sum of exactly ( k − 1) / 2 distinct pairs. Note that this is not p ossible if k is ev en. F or the case k o dd, for each i there is a matc hing { ( r i , s i ) } i ∈ [( k − 1) / 2] b et w een elemen ts in { a 1 , . . . , a k } \ a i suc h that a r j + a s j = a i . By considering the relation of the non-inv olution elemen t a t and its matched elemen t a t ′ and a i so that (so a t + a t ′ = a i with ( t, t ′ ) = ( r j , s j ) for some j ∈ [( k − 1) / 2]), and the relation a t ′ = a i + a i ′ for the i ′ -th elemen t matc hing the i -th, w e conclude that a t + a i ′ = 0. Since a t w as a non-inv olution elemen t, this prov es that there exists a non-in v olution a t ∈ A so that − a t ∈ A . No w, assuming that a i , − a i ∈ A with a i  = − a i , w e pro v e that 2 a j ∈ A ∪ { 0 } for all a j  = ± a i . Indeed, b oth a j + a i and a j − a i ∈ A ∪ { 0 } and, since these elemen ts are distinct, also ( a j + a i ) + ( a j − a i ) = 2 a j ∈ A ∪ { 0 } , and claim follo ws. Now, since 2 a j = a j + a j ∈ A ∪ { 0 } , it follo ws that, for all a j  = ± a i , the sets { 0 + a j , a 1 + a j , . . . , a k + a j } ⊆ A ∪ { 0 } . Since b oth these sets hav e cardinalit y k + 1, we m ust hav e that { 0 + a j , a 1 + a j , . . . , a k + a j } = A ∪ { 0 } and hence − a j ∈ A for all j (also when j = i ). Finally , w e consider 2 a i and wan t to see that 2 a i ∈ A ∪ { 0 } . W e assume a 1  = ± a i , and w e recall that we hav e { 0 + a 1 , a 1 + a 1 , . . . , a k + a 1 } = A ∪ { 0 } . Therefore, there exists j suc h that a i = a 1 + a j . If a j = a i , then we hav e a 1 = 0 a con tradiction, and if a j = − a i then a 1 = 2 a i as desired. If a j  = ± a i , then this means that 2 a i = a i + a i = ( a j + a 1 ) + a i = a j + ( a 1 + a i ) . Note that there exists an index h suc h that a 1 + a i = a i + a 1 = a h ∈ A which implies that 2 a i = a j + ( a 1 + a i ) = a j + a h = a h + a j ∈ A ∪ { 0 } where the last inclusion holds since a j  = ± a i . W e also note that the same argumen t (exc hang- ing a i b y − a i ) applies for 2( − a i ) = − 2 a i ∈ A . W e hav e pro v ed that 2 a j ∈ A ∪ { 0 } for all j (also when considering ± a i ). It follo ws that A ∪ { 0 } is b oth inv erse-closed and closed under the sums of its elemen ts and th us is a subgroup. □ Corollary 2.5 (Recursive Lemma) . L et G b e an ab elian gr oup, and let A = { a 1 , . . . , a k } ⊆ G \ { 0 } b e a set of k elements. Then we either se e that A is se quenc e able or that ther e exist distinct { i, j } such that a i + a j ∈ A ∪ { 0 } . 6 GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS Pr o of. Let us assume that a i + a j ∈ A ∪ { 0 } for any distinct i, j . Then, due to Theorem 2.4, w e either hav e that k = 1 in which case A is trivially sequenceable or that A = H \ { 0 } for some subgroup H of G . In the latter case w e hav e that A is sequenceable due to Theorem 1.2 of [2]. □ 3. Code description W e describe the algorithm used to establish Propositions 4.1 and 4.2. The method is a proof b y contradiction: we assume that A = { a 1 , . . . , a k } ⊆ G \ { 0 } is a c ounter example , meaning that every ordering of A has at least one forbidden collision among its partial sums. P artial sums and incidence v ectors. Let x = ( x 1 , . . . , x k ) b e an ordering of A and define partial sums y 0 = 0 , y i = i X j =1 x j (1 ≤ i ≤ k ) . A collision y s = y t with 0 ≤ s < t ≤ k is equiv alent to a consecutiv e zero-sum blo ck (1) x s +1 + x s +2 + · · · + x t = 0 . Fix a lab eling A = { a 1 , . . . , a k } and write the ordering as a permutation ω ∈ Sym( k ), so that x r = a ω ( r ) . Given an interv al of p ositions [ s + 1 , t ], we enco de which lab els app ear in that blo c k b y the incidenc e ve ctor v ∈ { 0 , 1 } k , defined by v i = 1 if a i o ccurs somewhere in the blo c k and v i = 0 otherwise. Then (1) b ecomes the lab el-based relation (2) k X i =1 v i a i = 0 . Along an y branch of the searc h tree we collect these v ectors as rows of a binary matrix C ∈ { 0 , 1 } m × k , where m is the num b er of recorded interv als on that branc h. An y integer com bination of the ro ws of C gives a further linear relation among the a i ’s. Searc h tree structure. A no de of the search tree is a pair ( ω , C ) consisting of a curren t ordering ω and the constrain t matrix C collected along the branc h. The ro ot is (id , ∅ ), where id = (1 , 2 , . . . , k ). T o generate children of a no de ( ω , C ), the algorithm considers every in terv al of p ositions [ i, j ] (with p ositions indexed from 0 to k − 1) satisfying 1 ≤ i < j ≤ k and j − i ≤  k 2  . The length restriction j − i ≤ ⌊ k / 2 ⌋ is used in the zero-sum case; for general (non-zero-sum) subsets we exclude only the full in terv al [0 , k − 1]. F or each admissible interv al [ i, j ] the algorithm: • builds its incidence v ector v ∈ { 0 , 1 } k b y setting v ω ( ℓ ) − 1 = 1 for ℓ ∈ [ i, j ] and 0 elsewhere; • skips this interv al if v already appears as a row of C (adding a duplicate row gives no new information); • p erforms one adjacent swap at a boundary of [ i, j ] to pro duce a new ordering ω ′ : if i ≥ 1, sw ap p ositions i and i − 1 (left b oundary mo v e), and if i = 0, sw ap p os itions j and j + 1 (right b oundary mov e); • creates the c hild no de ( ω ′ , C ∪ { v } ). GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS 7 Eac h child receiv es its o wn copy of the augmen ted matrix, so differen t branches are indep en- den t. The tree is explored breadth-first (BFS). A no de is not expanded further once a terminal certificate (describ ed next) is detected. T erminal certificates. At eac h no de ( ω , C ) the algorithm chec ks whether the row span of C o v er Q contains one of a small list of vectors. The chec k is alwa ys the same: for a candidate u ∈ Q k , compare rank( C ) and rank  C u  . If the tw o ranks coincide, then u lies in the ro w span of C and a con tradiction is obtained. • Zer o-element c ertific ate. If e i lies in the row span of C , then clearing denominators giv es an integer com bination of recorded relations equal to a i = 0, contradicting A ⊆ G \ { 0 } . • Equality c ertific ate. If e i − e j lies in the ro w span of C for some i  = j , then we obtain a i = a j , contradicting that the elements of A are distinct. • Compr ession c ertific ate. By the Recursive Lemma there exist a 1 , a 2 ∈ A such that A ′ := ( A \ { a 1 , a 2 } ) ∪ { a 1 + a 2 } has size k − 1 and admits a sequencing x 1 , . . . , x k − 1 . This fixed sequencing implies that certain consecutiv e blo c ks in A ′ are not zero-sum. These forbidden blocks can be listed as a finite set of binary v ectors initial cons ⊂ { 0 , 1 } k . If an y w ∈ initial cons lies in the row span of C , then the counterexample assumption forces a relation P i w i a i = 0 of a type that cannot o ccur in the compressed sequencing, giving a contradiction. All three chec ks are implemen ted using the same rank test, computed with standard nu- merical linear algebra ( numpy.linalg.matrix rank ). Correctness and completeness. The algorithm is correct in the following sense: when a terminal certificate is found, it is a purely linear consequence of the recorded relations (2), so it yields a con tradiction indep endently of the sp ecific abelian group G . It is complete in the follo wing sense: if the BFS exploration finishes with every branch either reac hing a terminal certificate or ha ving no children, then the counterexample assumption cannot b e sustained. In particular, no set A of the giv en size can b e a counterexample, so ev ery such set is sequenceable. Complexit y and implemen tation. A t depth d a no de has m = d recorded rows. T he n um b er of candidate interv als is at most  k 2  , reduced b y the length restriction and b y the deduplication c hec k. The rank test costs O ( m 2 k ) p er candidate, so a rough p er-no de b ound is O ( k 2 m 2 k ) = O ( m 2 k 3 ). F or the v alues of k considered here ( k ≤ 23), terminal certificates are t ypically reac hed after very few levels, so the practical running time is m uc h smaller than w orst-case b ounds. Example for k = 3 . W e give a minimal example where a con tradiction is obtained after recording only t w o interv al constrain ts. Let A = { a 1 , a 2 , a 3 } ⊆ G \ { 0 } with pairwise distinct elemen ts. Start from the ro ot no de with ordering ω = (1 , 2 , 3) , C = ∅ , and assume (as in the pro of by contradiction) that every ordering of A con tains a forbidden consecutiv e zero-sum block. 8 GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS Cho ose the in terv al [ i, j ] = [0 , 1] in the ordering (1 , 2 , 3). Its incidence v ector is v (1) = (1 , 1 , 0) , enco ding the relation a 1 + a 2 = 0. Since i = 0, the algorithm p erforms the right b oundary mo v e and sw aps p ositions j and j + 1, i.e. p ositions 1 and 2. This pro duces the new ordering ω ′ = (1 , 3 , 2) , and the child no de has constrain t matrix C = { v (1) } . F rom this child, choose the interv al [1 , 2] in the ordering (1 , 3 , 2). The lab els in these p ositions are { 3 , 2 } , so the incidence vector is v (2) = (0 , 1 , 1) , enco ding the relation a 2 + a 3 = 0. Now the matrix of recorded constrain ts is C = { (1 , 1 , 0) , (0 , 1 , 1) } . A t this point w e obtain a terminal certificate of e quality type, since v (1) − v (2) = (1 , 1 , 0) − (0 , 1 , 1) = (1 , 0 , − 1) = e 1 − e 3 . Equiv alently , e 1 − e 3 lies in the ro w span of C , so the algorithm stops. The corresp onding con tradiction in G is ( a 1 + a 2 ) − ( a 2 + a 3 ) = 0 = ⇒ a 1 − a 3 = 0 = ⇒ a 1 = a 3 , whic h contradicts the assumption that the elemen ts of A are distinct. 4. Conclusion By running the code 1 of the previous section w e obtain the follo wing statement. Prop osition 4.1. L et A ′ b e a subset of c ar dinality k − 1 c onsisting of nonzer o elements, of an ab elian gr oup G , summing to zer o, and supp ose A ′ admits a se quencing x 1 , x 2 , . . . , x k − 1 . Given two elements a 1 , a 2 such that A = { a 1 , a 2 , x 2 , x 3 , . . . , x k − 1 } is a set of c ar dinality k of nonzer o elements satisfying a 1 + a 2 = x 1 , we c onclude that A admits a se quencing in the fol lowing c ases: • k ≤ 22 , under the assumption that P a ∈ A a = 0 ; • k ≤ 23 , under the additional assumption that A c ontains no p air of the form { x, − x } . Therefore, by virtue of the Recursiv e Lemma Theorems 1.5 and 1.6 follow readily . In the case where A is not zero-sum, w e proceed analogously; ho wev er, in this case, w e can not assume that the merged elemen ts sum to x 1 since the ordering is not cyclic and w e ha v e to consider k − 1 cases. An yw ay , b y means of the computational search, w e obtain the follo wing prop osition. 1 All the computations were implemented in Python and run on a lo cal server (with 2 TB of RAM and 128 AMD CPU cores). All code used in this w ork is op enly av ailable at the following repository: 10.5281/zen- o do.18997904. GRAHAM CONJECTURE ON SMALL SETS IN ABELIAN GROUPS 9 Prop osition 4.2. L et A ′ b e a subset of c ar dinality k − 1 c onsisting of nonzer o elements, of an ab elian gr oup G and supp ose A ′ admits a se quencing x 1 , x 2 , . . . , x k − 1 . Given two elements a 1 , a 2 such that A = { a 1 , a 2 , x 1 , x 2 , . . . , x i − 1 , x i +1 , . . . , x k − 1 } is a set of c ar dinality k of nonzer o elements satisfying a 1 + a 2 = x i for some i in [1 , k − 1] , we c onclude that A admits a se quencing whenever k ≤ 20 . Therefore, also in this case, by the Recursive Lemma Theorem 1.4 follows. T o stress the effectiveness of our recursiv e approach, it is worth noting that a pure tree- algorithm (i.e. considering only the first t w o terminal certificates), without the Recursive Lemma, is limited to sets of size | A | ≤ 10, consistently with the current literature (see [3], whic h arrives at size 9). A cknowledgements The first and second authors w ere partially supp orted b y INdAM–GNSAGA. The last author has b een supp orted by the I+D+i pro ject PID2023-147202NB-I00 funded b y MI- CIU/AEI/10.13039/501100011033. References [1] N. Alon, Combinatorial Nullstellensatz, Combin. Pr ob ab. Comput. 8 (1999), 7–29. [2] B. Alspach, D. 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(Simone Costa) DICA T AM, Universit ` a degli Studi di Brescia Email address : simone.costa@unibs.it (Stefano Della Fiore) DI I, Universit ` a degli Studi di Brescia Email address : stefano.dellafiore@unibs.it (Mattia F on tana) Dep ar tment of Engineering and Sciences, Universit as Mer ca torum Email address : mattia.fontana@studenti.unimercatorum.it (Llu ´ ıs V ena) Dep ar tment of Ma thema tics, Universit a t Polit ` ecnica de Ca t aluny a Email address : lluis.vena@upc.edu