#A105 INTEGERS 25 (2025)
FROM SCHUR’S THEOREM TO THE PAIRWISE SUM THEOREM
Mau o Di Nasso
Dipa imen o di Ma ema ica, Uni e si `a di Pisa, I aly
[email p o ec ed]
Renling Jin
Depa men o Ma hema ics, College o Cha les on, Sou h Ca olina
[email p o ec ed]
Recei ed: 9/13/25, Accep ed: 11/3/25, Published: 11/25/25
Abs ac
We show ha by only assuming Schu ’s Theo em and he exis ence o a non-
p incipal ul a il e , one can di ec ly p o e ha in e e y ini e colo ing o N he e
exis in ini e disjoin se s A, B such ha all elemen s o A∪B∪(A+B) a e monoch o-
ma ic. This gi es a pa ial answe o a ques ion posed by N. Hindman, I. Leade ,
and D. S auss in 2003. In he las sec ion we p opose a o maliza ion o ha open
ques ion in pu ely opological e ms.
1. In oduc ion
In hei 2003 a icle [7] en i led “Open p oblems in pa i ion egula i y,” N. Hind-
man, I. Leade , and D. S auss compiled a lis o 13 open p oblems conside ed o be
he mos ele an in ha ield a he ime. A e mo e han wen y yea s, ques ions
enume a ed he e as Ques ions 5, 6, 7, 8, 9, 10, and 13, ha e now been sol ed.
Speci ically, Ques ions 5 and 6 ha e been se led wi h posi i e answe s in [1] by B.
Ba be , N. Hindman, and I. Leade , and Ques ions 7, 8, and 9 ha e been se led
wi h nega i e answe s in [2] by B. Ba be and I. Leade . Besides, I. Leade and
P.A. Russell ga e a nega i e answe o Ques ion 10 in he pape [9];1and inally,
he conjec u e o Ques ion 13 was ecen ly p o ed o be ue by Z. Zelenyk in [10].
He e we ocus on Open Ques ion 12 (see below), which is abou he s eng h o a
educed e sion o Hindman’s Theo em whe e only pai wise sums om an in ini e
sequence a e conside ed.
DOI: 10.5281/zenodo.17711561
1This pape [9] is ac ually e e enced in he open p oblems pape [7] as “manusc ip ”, bu
be o e publica ion he au ho s ound a coun e example and included i in he inal e sion.
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Be o e p oceeding, le us ecall some basic e minology. A ini e colo ing o a
se Xis a ini e pa i ion X=C1∪. . . ∪C whe e he pieces Cia e called colo s.
A se Ais monoch oma ic wi h espec o a gi en ini e colo ing i all elemen s
o Abelong o he same colo , i.e.,A⊆Ci o some i. A amily P⊆ P(X) is
called weakly pa i ion egula on Xi o e e y ini e colo ing o X he e exis s
A∈Pwhich is monoch oma ic. Fo example, Schu ’s Theo em s a es ha he
amily {{a, b, a +b} | a, b ∈N}is weakly pa i ion egula on N.
Le us ecall he Fini e Sums Theo em, a co ne s one o combina o ics p o ed
by N. Hindman in 1974.
Theo em 1.1 (Fini e Sums Theo em [6]).Fo e e y ini e colo ing o he na u al
numbe s he e exis s an in ini e se Asuch ha i s se o ini e sums
FS(A) := (X
a∈F
a| ∅ =F⊂A ini e)
is monoch oma ic.
Sho ly a e Hindman’s a icle was published, Gal in and Glaze ound a di -
e en p oo using “idempo en ul a il e s”; his p oo pa ed he way o an en i e
ield o esea ch based on he algeb aic p ope ies o he space o ul a il e s βN(see
he comp ehensi e monog aph [8]).
The ollowing p oblem was posed in [7] as Open Ques ion 12, wi h he commen :
“I seems uly ema kable ha his can be unknown”.
(Q12) Is he e a p oo ha whene e Nis ini ely colou ed he e is a 1-1 sequence
x1, x2, . . . such ha all xiand all xi+xj(i=j) ha e he same colou , ha
does no also p o e he Fini e Sums Theo em?
Fo con enience, le us gi e a name o he p ope y conside ed in he abo e
ques ion.
Theo em 1.2 (Pai wise Sum Theo em).Fo e e y ini e colo ing o he na u al
numbe s he e exis s an in ini e Asuch ha he se o pai wise sums
FS≤2(A) = A∪ {a+a′|a, a′∈A, a =a′}
is monoch oma ic.
We will show ha by only using Schu ’s Theo em and he exis ence o a non-
p incipal ul a il e on N, one can p o e he ollowing weake e sion o he Pai wise
Sum Theo em, hus p o iding a pa ial answe o (Q12).
Theo em 2.2.Fo e e y ini e colo ing o he na u al numbe s he e exis in ini e
disjoin se s A, B such ha A∪B∪(A+B)is monoch oma ic.
We obse e ha (Q12), as o mula ed in [7], is somewha ague and ambigu-
ous. In he las sec ion we p opose a possible igo ous o maliza ion as a pu ely
opological p ope y o he space βN2.
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2. The P oo
Le us i s ecall a ew no ions and ac s abou ul a il e s. The enso p oduc
U ⊗ V be ween ul a il e s on Nis de ined by se ing, o e e y X⊆N×N,
X∈ U ⊗ V i and only i {n∈N|Xn∈ V} ∈ U
whe e Xn={m∈N|(n, m)∈X}is he e ical n- ibe o X. The pseudo-sum
U ⊕ V is he ul a il e on Nde ined by se ing, o e e y A⊆N,
A∈ U ⊕ V i and only i {n∈N|A−n∈V}∈U
whe e A−n:= {m∈N|m+n∈A}. Obse e ha U ⊕ V = Sum(U ⊗ V) is he
image ul a il e o he enso p oduc unde he sum unc ion Sum(n, m)=n+m.
Recall ha i Uis an ul a il e on a se Iand :I→J, he image ul a il e (U)
on Jis de ined by se ing A∈ (U) i and only i −1(A)∈ U o all A⊆J.2
In ou p oo , we will use he ollowing combina o ial p ope y o enso p oduc s.
I is a pa icula case o a gene al esul p esen ed in [3]. Howe e , o comple eness,
we gi e he e a sel -con ained p oo .
Lemma 2.1. Fo e e y X⊆N×N, he ollowing condi ions a e equi alen .
1. The e exis disjoin inc easing sequences (an)n∈Nand (bn)n∈Nsuch ha
{(ai, bj)|i≤j} ∪ {(bj, ai)|j < i}⊆X.
2. The e exis non-p incipal ul a il e s V1,V2on Nsuch ha
X∈(V1⊗ V2)∩(V2⊗ V1).
P oo . (1) ⇒(2). Le A={an|n∈N}and B={bn|n∈N}. We obse e
ha he amily F1:= {A}∪{Xbj|j∈N} ∪ {[n, ∞]|n∈N}has he ini e
in e sec ion p ope y; indeed, ai∈A∩Xb1∩...∩Xbk∩[n, ∞] o e e y i > k, n.
Simila ly, he amily F2:= {B} ∪ {Xai|i∈N} ∪ {[n, ∞]|n∈N}also has he
ini e in e sec ion p ope y. So, we can pick non-p incipal ul a il e s V1⊇ F1and
V2⊇ F2. No e ha he se Y1:= {(ai, bj)|i≤j} ∈ V1⊗ V2because o e e y
as∈A he e ical ibe (Y1)as={bj|j≥s}=B∩[as,∞]∈ V2. Simila ly,
Y2:= {(bj, ai)|i≤j} ∈ V2⊗ V1because o e e y b ∈B he e ical ibe
(Y2)b ∈ V1. We conclude ha X⊇Y1∪Y2∈(V1⊗ V2)∩(V2⊗ V1), as desi ed.
(2) ⇒(1). Le X(1) := {n|Xn∈ V1}and X(2) := {n|Xn∈ V2}. By
de ini ion, X∈ V1⊗ V2⇔X(2) ∈ V1and X∈ V2⊗ V1⇔X(1) ∈ V2. Pick
2The s anda d e e ence o pseudo-sums o ul a il e s, and mo e gene ally o he algeb a on
he space o ul a il e s on a se Sas de e mined by an a bi a y associa i e ope a ion on S, is
Hindman-S auss’ book [8]. Fo mo e in o ma ion on enso p oduc s, see [8, §11.1]; see also [3]
whe e hei combina o ial p ope ies a e in es iga ed.
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a1∈X(2); hen Xa1∈ V2. Pick b1∈X(1) ∩Xa1∈ V2wi h b1> a1( his is
possible because V2is non-p incipal); hen (a1, b1)∈Xand Xb1∈ V1. Induc i ely,
pick an+1 ∈X(2) ∩Tn
i=1 Xbi∈ V1wi h an+1 > bn, so ha (bi, an+1)∈X o
all i= 1, . . . , n and Xan+1 ∈ V2. Then pick bn+1 ∈X(1) ∩Tn+1
i=1 Xai∈ V2wi h
bn+1 > an+1, so ha (ai, bn+1)∈X o all i= 1,...,n + 1 and Xbn+1 ∈ V1.
I is easily e i ied ha he inc easing sequences (an) and (bn) sa is y he desi ed
p ope ies.
We a e inally eady o gi e a p oo o he Pai wise Sum Theo em.
Theo em 2.2. Fo e e y ini e colo ing o he na u al numbe s he e exis in ini e
disjoin se s A, B such ha A∪B∪(A+B)is monoch oma ic.
P oo . Le Ube a non-p incipal ul a il e on N. Fo n∈N, deno e by Un⊕ he
i e a ed sum o he ul a il e Uwi h i sel , i.e.,U1⊕=U, and induc i ely U(n+1)⊕=
Un⊕⊕ U. We obse e ha Un⊕⊕ Um⊕=U(n+m)⊕.
Gi en a ini e colo ing N=C1∪. . . ∪C , conside he colo ing N=D1∪. . . ∪D
whe e we pu n∈Dii and only i Ci∈ Un⊕. By Schu ’s Theo em he e exis s
a colo Diand a pai a=bsuch ha a, b, a +b∈Di. This means ha he colo
C=Ciis a membe o V1∩V2∩W, whe e V1:= Ua⊕,V2:= Ub⊕, and W:= U(a+b)⊕.
We now obse e ha he image ul a il e s π1(V1⊗ V2) = π2(V2⊗ V1)=V1and
π2(V1⊗ V2)=π1(V2⊗ V1)=V2whe e π1, π2:N×Na e he canonical p ojec ions.
We also obse e ha W=V1⊕ V2=V2⊕ V1= Sum(V1⊗ V2) = Sum(V2⊗ V1),
and so,
X:= π−1
1(C)∩π−1
2(C)∩Sum−1(C) =
={(a, b)∈C×C|a+b∈C}∈(V1⊗ V2)∩(V2⊗ V1).
Now apply Lemma 2.1 o X, and ob ain he exis ence o disjoin inc easing
sequences (an)n∈Nand (bn)n∈Nsuch ha
Γ:={(ai, bj)|i≤j} ∪ {(bj, ai)|j < i}⊆X.
Le A:= {an|n∈N}and B:= {bn|n∈N}. Then A=π1(Γ) ⊆π1(X)⊆C,
B=π2(Γ) ⊆π2(X)⊆C, and A+B= Sum(Γ) ⊆Sum(X)⊆C, as desi ed.
3. A Topological Fo maliza ion
As al eady ema ked in he in oduc ion, Open Ques ion (Q12) as o mula ed in [7]
is somewha ague and ambiguous. He e we p opose a o maliza ion in opological
e ms.
As i is well-known, he e is a close connec ion be ween weak pa i ion egula i y
and ul a il e s, g ounding on he ollowing ac .
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P oposi ion 3.1 ([8, Thm. 5.7]).Le F ⊆ P(S). The ollowing p ope ies a e
equi alen :
1. Fis weakly pa i ion egula on S.
2. The e exis s an ul a il e Uon Ssuch ha o e e y A∈ U he e exis s F∈ F
wi h F⊆A.
An ul a il e Uas abo e is called a PR-wi ness o he amily F.
No a ion 3.2. Following a common use, gi en se s o na u al numbe s
A={an|n∈N}and B={bn|n∈N}
whe e he sequences (an) and (bn) a e inc easing, we w i e
A⊕B={ai+bj|i < j}.
In pa icula , A⊕Acon ains all pai wise sums o dis inc elemen s o A.3
No ice ha Hindman’s Theo em s a es ha H:= {FS(X)|X⊆Nin ini e}is
a weakly pa i ion egula amily; and simila ly, he Pai wise Sum Theo em s a es
ha S2={B∪(B⊕B)|B⊆Nin ini e}is a weakly pa i ion egula amily.
One can o malize he Open Ques ion (Q12) as a p ecise ma hema ical s a emen
in e ms o wi ness ul a il e s, as ollows:
(†)Is he e an ul a il e U ha is a PR-wi ness o S2bu no a PR-wi ness o
H?
As we will show below, p ope y (†) can be e o mula ed in pu ely opological
e ms wi hin he S one–ˇ
Cech compac i ica ion βN2o he disc e e space N2. Recall
ha βN2is usually ep esen ed as he space o ul a il e s on N2whe e a base o
(cl)open se s is gi en by he amily o se s o he o m OX:= {W ∈ βN2|X∈ W}
(see [8, Sec ion 3.2]). Recall he ollowing ac .
P oposi ion 3.3 ([8, Lemma 5.19]).An ul a il e is a PR-wi ness o he amily H
i and only i i belongs o he opological closu e o he se o idempo en s:
H={U ∈ βN| U ⊕ U =U}.
In he ecen pape [4], he class o wi nesses o he PR p ope y gi en by Ram-
sey’s Theo em on pai s was in oduced and s udied.
3We obse e ha he same ⊕symbol is also used o he “pseudo-sum” ope a ion be ween
ul a il e s on N; howe e , he e his should no cause misunde s andings.
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De ini ion 3.4. An ul a il e Won N2isaRamsey’s wi ness i o e e y X∈ W
he e exis s an in ini e Hsuch ha
[H]2={(h, h′)∈H×H|h < h′}⊆X.
The ollowing ac was al eady poin ed ou in [4]. Fo comple eness, we gi e a
p oo he e.
P oposi ion 3.5. An ul a il e W ∈ βN2is a Ramsey’s wi ness i and only i i
belongs o he closu e To he se o enso powe s T:= {U ⊗ U | U ∈ βN N}.
P oo . Le W ∈ T, and le X∈ W. Then X∈ U ⊗U o some non-p incipal U. Now
we obse e ha e e y such enso powe is a Ramsey’s wi ness. Indeed, X∈ U ⊗ U
i and only i XU:= {n|Xn∈U}∈U. Pick h1∈XUand, induc i ely, pick
hn+1 ∈XU∩Xh1∩. . . ∩Xhn∈ U. Since Uis non-p incipal we can pick hn+1 > hn.
Then H={hn|n∈N}is he desi ed in ini e homogeneous se [H]2⊆X.
Con e sely, le H={h1< . . . < hn< . . .}be an in ini e se wi h [H]2⊆X.
The amily F:= {{hm|m≥n} | n∈N}has he ini e in e sec ion p ope y, and
i is easily e i ied ha e e y (necessa ily non-p incipal) ul a il e U ⊇ F is such
ha [H]2∈ U ⊗ U, and hence X∈ U ⊗ U.
We now obse e he ollowing p ope y ha connec s Ramsey’s wi nesses wi h
he Pai wise Sum Theo em.
P oposi ion 3.6. Uis a PR-wi ness o he amily S2i and only i he e exis s a
Ramsey’s wi ness Wsuch ha U=π1(W)=π2(W)=Sum(W).
P oo . Le A∈ U =π1(W)=π2(W) = Sum(W). Then
XA:= π−1
1(A)∩π−1
2(A)∩Sum−1(A)={(a, a′)∈A×A|a+a′∈A} ∈ W.
Pick an in ini e B={b1< . . . < bn< . . .}wi h [B]2⊆XA. Then i is easily
e i ied om he de ini ions ha B∪(B⊕B)⊆A.
Fo he con e se implica ion, obse e ha an ul a il e Won N2has he p ope y
π1(W) = π2(W) = Sum(W) = Ui and only i he p eimages π−1
1(A), π−1
2(A),
and Sum−1(A) belong o W o e e y A∈ U; equi alen ly, i and only i ΓA:=
(A×A)∩Sum−1(A)∈ W o e e y A∈ U. Obse e also ha an ul a il e Won
N2is a Ramsey’s wi ness i and only i Wex ends he ollowing amily:
R={X⊆N2|[H]2⊆ Xc o e e y in ini e H}.
Thus we each he hesis i we show ha he e exis s an ul a il e W ha ex ends
he amily R∪{ΓA|A∈ U}, and his is equi alen o ha ing R∪{ΓA|A∈ U}
sa is y he ini e in e sec ion p ope y. Assume owa ds a con adic ion ha he e
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a e X1, . . . , Xs∈ R and A1, . . . , A ∈ U such ha Ts
i=1 Xi∩T
j=1 ΓAj=∅. I
A:= A1∩. . . ∩A hen we ha e
(A×A)∩Sum−1(A) =
j=1
ΓAj⊆(X1)c∪. . . ∪(Xs)c.
Since A∈ U we can pick an in ini e B={b1< . . . < bn< . . .}such ha B∪
(B⊕B)⊆A. This means ha [B]2⊆(A×A)∩Sum−1(A)⊆(X1)c∪. . . ∪(Xs)c.
Finally, conside he ini e colo ing [N]2=C1∪...∪Cswhe e (n, m)∈Ci⇔iis
he leas index such ha (bn, bm)∈(Xi)c. By Ramsey’s Theo em he e exis s an
in ini e Hsuch ha i s pai s [H]2⊆Cia e monoch oma ic. Bu hen we would
ha e [{bh|h∈H}]2⊆(Xi)c, a con adic ion.
Pu ing all o he abo e oge he , we can inally gi e a p ecise o maliza ion o
Open Ques ion 12 om [7] as he ollowing p ope y o he opological space βN2,
ob ained as a e o mula ion o he p e ious (†).
(Q12) Conside he ollowing subspaces o βN2:
–T:= {U ⊗ U | U ∈ βN N}.
–S:= {W ∈ βN2|π1(W)=π2(W) = Sum(W)},
Is i ue ha Sum T∩S=Sum (T∩S)?
Acknowledgemen . M. Di Nasso is suppo ed by he I alian esea ch p ojec
PRIN 2022: “Logical me hods in combina o ics”, 2022BXH4R5, MIUR (I alian
Minis y o Uni e si y and Resea ch), and is a membe o he INdAM esea ch
g oup GNSAGA. R. Jin is suppo ed by Simons Founda ion (g an numbe 513023).
Re e ences
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Se . A 120 (2013), 1590-1599.
[2] B. Ba be and I. Leade , Pa i ion egula i y wi h cong uence condi ions, J. Comb. 4(2013),
293-297.
[3] M. Di Nasso, The magic o enso p oduc s o ul a il e s, p ep in , A Xi :2506.14344.
[4] M. Di Nasso, L. Lupe i Baglini, M. Mamino, R. Mennuni, and M. Ragos a, Ramsey’s wi nesses,
p ep in , A Xi :2503.09246.
[5] M. Di Nasso and R. Jin, Founda ions o i e a ed s a maps and hei use in combina o ics,
Ann. Pu e Appl. Logic 176 (2025), 103511.
INTEGERS: 25 (2025) 8
[6] N. Hindman, Fini e sums om sequences wi hin cells o a pa i ion o N,J. Comb. Theo y
Se . A 17 (1974), 1-11.
[7] N. Hindman, I. Leade , and D. S auss, Open p oblems in pa i ion egula i y, Combina o ics,
P obabili y, and Compu ing 12 (2003), 571-583.
[8] N. Hindman and D. S auss, Algeb a in he S one- ˇ
Cech compac i ica ion: Theo y and Appli-
ca ions, 2nd ed., Wal e de G uy e , Be lin, 2012.
[9] I. Leade and P.A. Russell, Independence o pa i ion egula equa ions, J. Comb. Theo y
Se . A 114 (2007), 825-839.
[10] Y. Zelenyuk, Elemen s o o de 2 in βN,Fund. Ma h. 252 (2021), 355-360.
