Finite sums of arithmetic progressions

TL;DR

This paper provides a combinatorial proof for a generalized theorem related to arithmetic progressions and Hindman's theorem, along with bounds for finite cases.

Contribution

It introduces a new combinatorial proof for a two-fold generalization of classical theorems and establishes tower bounds for finite versions.

Findings

Purely combinatorial proof of the generalization

Tower bounds for finite cases

Extension of van der Waerden-Brauer and Hindman's theorems

Abstract

We give a purely combinatorial proof for a two-fold generalization of van der Waerden-Brauer's theorem and Hindman's theorem. We also give tower bounds for a finite version of it.

Equations

$$\underset{i\in B}{⨁}{P}_i=P_1\oplus P_2\oplus \cdots \oplus P_m$$

$$\underset{i\in B}{⨁}{P}_i[s]=\sum _{i\in B}{P}_i[s].$$

$$\mathbf{c}(d)=\mathbf{c}(a)=\mathbf{c}(a+d)=\cdots =\mathbf{c}(a+(l-1)d).$$

$$b_1=\sum _{i\in B_1}{a}_i,b_2=\sum _{i\in B_2}{a}_i,\dots ,b_m=\sum _{i\in B_m}{a}_i,\dots$$

$$\mathbf{c}\big{(}\displaystyle\sum_{i\in C}a_{i}\big{)}=\gamma.$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in C}Q_{i}\,[s]\big{)}=\gamma.$$

$$P_i^0=\{a_i,a_i+1,\dots ,a_i+(n-1)\}.$$

$$P_1^0\prec P_2^0\prec P_3^0\prec \cdots$$

$$\underset{i\in C_1}{⨁}{P}_i^0\prec \underset{i\in C_2}{⨁}{P}_i^0.$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in C}P_{i}^{k}\,[s]\big{)}=\alpha_{s}^{k},$$

$$\underset{i\in C_1}{⨁}{P}_i^k\prec \underset{i\in C_2}{⨁}{P}_i^k.$$

$$P_1^k[k+1]<P_2^k[k+1]<\cdots <P_m^k[k+1]<\cdots \cdot$$

$$b_1=\sum _{i\in B_1}{P}_i^k[k+1],b_2=\sum _{i\in B_2}{P}_i^k[k+1],\dots ,b_m=\sum _{i\in B_m}{P}_i^k[k+1],\dots$$

$$P_1^{k+1}=\underset{i\in B_1}{⨁}{P}_i^k,P_2^{k+1}=\underset{i\in B_2}{⨁}{P}_i^k,\dots ,P_m^{k+1}=\underset{i\in B_m}{⨁}{P}_i^k,\dots .$$

$${\alpha }_1^{k+1}={\alpha }_1^k,{\alpha }_2^{k+1}={\alpha }_2^k,\dots ,{\alpha }_k^{k+1}={\alpha }_k^k,{\alpha }_{k+1}^{k+1}=\alpha .$$

$$\underset{i\in C}{⨁}{P}_i^{k+1}[s]=\underset{i\in C}{⨁}\underset{j\in B_i}{⨁}{P}_j^k[s]=\underset{i\in D}{⨁}{P}_i^k[s],$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in D}P_{i}^{k}\,[s]\big{)}=\alpha_{s}^{k}=\alpha_{s}^{k+1}.$$

$$\underset{i\in C}{⨁}\underset{j\in B_i}{⨁}{P}_j^k[k+1]=\sum _{i\in C}\sum _{j\in B_i}{P}_j^k[k+1]=\sum _{i\in C}{b}_i\in FS(\{b_1,b_2,\dots \}),$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in C}\displaystyle\bigoplus_{j\in B_{i}}\!P_{j}^{k}\,[k+1]\big{)}=\mathbf{c}\big{(}\displaystyle\sum_{i\in C}b_{i}\big{)}=\alpha=\alpha_{k+1}^{k+1}.$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in C}P_{i}^{k+1}\,[s]\big{)}=\alpha_{s}^{k+1}$$

$$\underset{i\in C_1}{⨁}{P}_i^{k+1}\prec \underset{i\in C_2}{⨁}{P}_i^{k+1}$$

$$\underset{i\in C_1}{⨁}\underset{j\in B_i}{⨁}{P}_j^k\prec \underset{i\in C_2}{⨁}\underset{j\in B_i}{⨁}{P}_j^k.$$

$$\underset{i\in D_1}{⨁}{P}_i^k\prec \underset{i\in D_2}{⨁}{P}_i^k$$

$$\mathbf{c}(P_1^n[1])={\alpha }_1^n,\dots ,\mathbf{c}(P_1^n[n])={\alpha }_n^n.$$

$${\alpha }_a^n={\alpha }_{a+d}^n=\cdots ={\alpha }_{a+(l-1)d}^n=\gamma .$$

$$Q_{i}=\big{\{}P_{i}^{n}[a],P_{i}^{n}[a+d],\dots,P_{i}^{n}[a+(l-1)d]\big{\}}.$$

$$\mathbf{c}\big{(}\displaystyle\bigoplus_{i\in C}\!Q_{i}\,[s]\big{)}=\mathbf{c}\big{(}\displaystyle\sum_{i\in C}Q_{i}[s]\big{)}=\mathbf{c}\big{(}\displaystyle\sum_{i\in C}P_{i}^{n}[a+(s-1)d]\big{)}=\alpha^{n}_{a+(s-1)d}=\gamma.$$

$$\mathbf{c}\big{(}\!\displaystyle\bigoplus_{i\in C}\!Q_{i}\,[s]\big{)}=\mathbf{c}\big{(}\!\operatorname{add}\displaystyle\bigoplus_{i\in C}\!Q_{i}\big{)}=\gamma.$$

$$\mathrm{add}\underset{i\in C}{⨁}{P}_i^k=\underset{i\in C}{⨁}{P}_i^k[1].$$

$$\mathrm{add}\underset{i\in C}{⨁}{P}_i^0=\underset{i\in C}{⨁}{P}_i^0[2]-\underset{i\in C}{⨁}{P}_i^0[1]$$

Full text

1. Introduction

Let $l\geq 3$ be a positive integer and let $P=\{a_{1},\dots,a_{l}\}$ be an $l$-term arithmetic progression with $a_{1}<\dots<a_{l}$, we denote the $s$th term of $P$ by $P[s]=a_{s}$. Now let $P$ and $Q$ be two $l$-term arithmetic progressions, we define their pointwise sum (or briefly their sum) $P\oplus Q$ as the $l$-term arithmetic progression with $P\!\oplus\!Q\,[s]=P[s]+Q[s]$ for $1\leq s\leq l$. Hence for the $l$-term arithmetic progressions $P_{1},P_{2},\dots,P_{m}$, their finite sum $P_{1}\oplus P_{2}\oplus\dots\oplus P_{m}$ has unambiguous meaning. The following pleasant two-fold generalization of van der Waerden-Brauer’s theorem and Hindman’s theorem, can be deduced form either Furstenberg’s theorem ([3] Proposition 8.2.1) or Deuber-Hindman’s theorem [1].

For any positive integers $c$ and $l\geq 3$, if $\mathbb{N}$ is $c$-colored, then there exist a color $\gamma$ and infinitely many $l$-term arithmetic progressions $Q_{i}$, $i\in\mathbb{N}$ such that all of their finite sums (with no repetition) are monochromatic with the color $\gamma$ and all the common differences of the above finite sums have also the color $\gamma$ too.

It would be pleasant too if we have a purely combinatorial proof of such a statement avoiding topological dynamics as well as the theory of ultrafilters. In Theorems 3.1 and 3.2 of this paper we give such a proof. It is interesting to see whether the method of the proof can be generalized to give a combinatorial proof of Deuber-Hindman’s theorem [1]. We are also interested in a finite version of the above theorem. It is well known that through a compactness argument we can have a finite version. For instance we have the following theorem which is a two-fold generalization of van der Waerden’s theorem and a finite version of Hindman’s theorem.

*For positive integers $c,n$ and $l\geq 3$ there is a positive integer $m$ such that whenever $\{1,2,\dots,m\}$ is $c$-colored, then there exist $l$-term arithmetic progressions $P_{1},P_{2},\dots,P_{n}\subset\{1,2,\dots,m\}$ such that $\sum_{i=1}^{n}P_{i}[l]$ is not bigger than $m$ and all finite sums of $P_{i}$ (with no repetition) are monochromatic with the same color.

If we denote the least such $m$ by $f(l,n,c)$ then the proof given through the compactness argument does not give us upper bounds for $f(l,n,c)$. But it is not hard to see that the proof given for Theorem 3.1 can be made finitary (which may be regarded as an advantage of the proof over its counterparts using dynamical system or ultrafilters) to give us a primitive recursive upper bound for $f(l,n,c)$. To do so we use the finitary Hindman numbers $\textrm{Hind}(n,c)$ which is a tower function [2]. However due to its iterated use of the function $\textrm{Hind}(n,c)$, it gives us an upper bound belonging to the class of WOW functions [5]. In Theorem 4.1, we do a better job by giving a different proof which uses the function $\textrm{Hind}(n,c)$ just one time and thus obtaining tower bounds for $f(l,n,c)$. Also note that according to the Gowers elementary bounds for the van der Waerden theorem, we don’t worry about the van der Waerden part of the proof.

2. Preliminaries

Let’s fix some notations. For $n$ a positive integer put $[n]=\{1,2,\dots,n\}$. Let $S$ be an infinite set, we denote the collection of finite nonempty subsets of $S$ by $\mathcal{P}_{f}(S)$. For a finite set $A$, $\mathcal{P}^{+}(A)$ denotes the collection of nonempty subsets of $A$. Also $FS(S)$ will denote the set of all finite sums of elements of $S$ with no repetition. Let $A,B\in\mathcal{P}_{f}(\mathbb{N})$, by $A<B$ we mean that $\max A<\min B$. We also denote the common difference of the arithmetic progression $P$ by $\operatorname{add}{P}$. We use the following notation for finite sums of arithmetic progressions

[TABLE]

where $B=\{1,2,\dots,m\}$. Obviously we have

[TABLE]

We define a partial ordering between $l$-term arithmetic progressions by putting $P\prec Q$ whenever $P[s]<Q[s]$ for all $1\leq s\leq l$. Let’s state van der Waerden’s theorem and van der Waerden-Brauer’s theorem [5].

Theorem 2.1** (van der Waerden).**

For positive integers $c$ and $l\geq 3$ there is a positive integer $n$ such that whenever $[n]$ is $c$-colored, then there is a monochromatic $l$-term arithmetic progression $P\subseteq[n]$. We denote the least such $n$ by $W(l,c)$.

Theorem 2.2** (van der Waerden-Brauer).**

For positive integers $c$ and $l\geq 3$ there is a positive integer $n$ such that whenever $\mathbf{c}$ is a $c$-coloring of $[n]$, then there are $d,a,a+d,\dots,a+(l-1)d$ in $\{1,2,\dots,n\}$ such that

[TABLE]

We denote the least such $n$ by $WB(l,c)$.

We will use the following strong version of Hindman’s theorem [6].

Theorem 2.3**.**

Let $a_{1}<a_{2}<\dots<a_{m}<\dots$ be an infinite strictly increasing sequence of positive integers. Let $c$ be a positive integer and $FS(\{a_{1},a_{2},\dots\})$ be $c$-colored. Then there are $B_{1}<B_{2}<B_{3}<\dots$ in $\mathcal{P}_{f}(\mathbb{N})$ such that whenever

[TABLE]

then $FS(\{b_{1},b_{2},\dots\})$ is monochromatic.

We say that the two positive integers $a,b$ are power-disjoint, if the powers occurring in the expansions of $a,b$ in base $2$ are disjoint sets, more precisely if we write $a=2^{k_{1}}+\dots+2^{k_{m}}$ and $b=2^{l_{1}}+\dots+2^{l_{n}}$, then the two sets $\{k_{1},\dots,k_{m}\}$ and $\{l_{1},\dots,l_{n}\}$ are disjoint. We denote the set $\{k_{1},\dots,k_{m}\}$ by $\operatorname{pow_{2}}(a)$. We will use the following finitary version of Hindman’s theorem [2] which strengthens the Disjoint Unions Theorem. First we introduce a notation. If $T$ is a collection of pairwise disjoint sets, then $NU(T)$ will denote the set of non-empty unions of elements of $T$.

Theorem 2.4**.**

For positive integers $n,c$ there is a positive integer $m$ such that for any $m$-element set $A=\{a_{1},\dots,a_{m}\}$ of pairwise power-disjoint positive integers, whenever $\mathbf{c}$ is a $c$-coloring of $FS(A)$, then there exist $\gamma\in[c]$ and $B_{1},\dots,B_{n}$ in $\mathcal{P}^{+}([m])$ such that $B_{1}<\cdots<B_{n}$ and for all $C\in NU\{B_{1},\dots,B_{n}\}$ we have

[TABLE]

Moreover if $Hind(n,c)$ denotes the least such $m$, then $Hind(n,c)$ is a tower function.

3. Purely Combinatorial Proofs

In the following theorem we give a purely combinatorial proof of the two-fold generalization van der Waerden’s theorem and Hindman’s theorem mentioned in the introduction.

Theorem 3.1**.**

Let $c$ and $l\geq 3$ be positive integers. Let $\mathbf{c}$ be a $c$-coloring of $\mathbb{N}$, then there are $l$-term arithmetic progressions $Q_{1},Q_{2},Q_{3},\dots$ such that

• (i)

$Q_{1}\prec Q_{2}\prec Q_{3}\prec\cdots$,

• (ii)

there is $\gamma\in[c]$ such that for all $C\in\mathcal{P}_{f}(\mathbb{N})$ and all $s\in\{1,\dots,l\}$ we have

[TABLE]

Proof.

Let $n=W(l,c)$ and let $a_{1}<a_{2}<\cdots<a_{m}<\cdots$ be a strictly increasing sequence of positive integers with $a_{m+1}>a_{1}+\dots+a_{m}+mn$. For $i\in\mathbb{N}$ we put

[TABLE]

Obviously $P^{0}_{i}$ is an $n$-term arithmetic progression and we have

[TABLE]

In fact it is easily seen that for any $C_{1}<C_{2}$ in $\mathcal{P}_{f}(\mathbb{N})$ we have

[TABLE]

Now for $1\leq k\leq n$ we inductively define the $n$-term arithmetic progressions $P_{1}^{k},P_{2}^{k},P_{3}^{k},\ldots$ so that there are $\alpha_{1}^{k},\alpha_{2}^{k},\dots,\alpha_{k}^{k}\in[c]$ such that the following two conditions are satisfied

• (a)

for all $C\in\mathcal{P}_{f}(\mathbb{N})$ and all $s\in\{1,\dots,k\}$ we have

[TABLE]

• (b)

for all $C_{1}<C_{2}$ in $\mathcal{P}_{f}(\mathbb{N})$ we have

[TABLE]

Suppose we have defined $P_{1}^{k},P_{2}^{k},P_{3}^{k},\ldots$ with the above properties. We do the job for $k+1$. The second condition implies that

[TABLE]

Now by Hindman’s theorem there are $B_{1}<B_{2}<\cdots<B_{m}<\cdots$ in $\mathcal{P}_{f}(\mathbb{N})$ such that if we put

[TABLE]

then $\mathbf{c}$ has a constant value on $FS(\{b_{1},b_{2},\dots\})$, which we denote it by $\alpha$. Now we set

[TABLE]

as well as we set

[TABLE]

We check the conditions (a) and (b) for $k+1$. Let $C\in\mathcal{P}_{f}(\mathbb{N})$ and $1\leq s\leq k+1$, hence we have

[TABLE]

where $D=\bigcup_{i\in C}B_{i}$. Suppose $1\leq s\leq k$, from the induction hypothesis it follows that

[TABLE]

Also for $s=k+1$ we have

[TABLE]

which implies that

[TABLE]

Now putting (2) and (3) together we deduce

[TABLE]

for $1\leq s\leq k+1$. This finishes the proof of the condition (a). Now we turn to checking (b). Let $C_{1}<C_{2}$ be in $\mathcal{P}_{f}(\mathbb{N})$. We must show that

[TABLE]

which is equivalent to

[TABLE]

Letting $D_{1}=\bigcup_{i\in C_{1}}B_{i}$, $D_{2}=\bigcup_{i\in C_{2}}B_{i}$, we get $D_{1}<D_{2}$ and (4) becomes

[TABLE]

which is exactly our induction hypothesis. This proves the condition (b).

Now consider $P_{1}^{n}[1],P_{1}^{n}[2],\dots,P_{1}^{n}[n]$ and recall that $n=W(l,c)$. By construction we have

[TABLE]

Through induced coloring, it follows from van der Waerden’s theorem that there exist $\gamma\in[c]$ and positive integers $a,d$ such that

[TABLE]

We define the desire arithmetic progressions $Q_{i},i\in\mathbb{N}$ as follows

[TABLE]

It is easily seen by condition (b) that $Q_{1}\prec Q_{2}\prec Q_{3}\prec\cdots\,\cdot$ Also for all $C\in\mathcal{P}_{f}(\mathbb{N})$ and all $1\leq s\leq l$ we have

[TABLE]

This finishes the proof of Theorem 3.1. ∎

Now we turn to the two-fold generalization of van der Waerden-Brauer’s theorem and Hindman’s theorem.

Theorem 3.2**.**

Let $c$ and $l\geq 3$ be positive integers. Let $\mathbf{c}$ be a $c$-coloring of $\mathbb{N}$, then there are $l$-term arithmetic progressions $Q_{1},Q_{2},Q_{3},\dots$ such that

• (i)

$Q_{1}\prec Q_{2}\prec Q_{3}\prec\cdots$,

• (ii)

there is $\gamma\in[c]$ such that for all $C\in\mathcal{P}_{f}(\mathbb{N})$ and all $s\in\{1,\dots,l\}$ we have

[TABLE]

Proof.

We start with $n=WB(l,c)$ and a strictly increasing sequence of positive integers $a_{1}<a_{2}<\dots<a_{m}<\cdots$ with $a_{m+1}>n(a_{1}+\dots+a_{m})$. For $i\in\mathbb{N}$, We put $P^{0}_{i}=\{a_{i},a_{i}+a_{i},\dots,a_{i}+(n-1)a_{i}\}$. In this case for all $1\leq k\leq n$ and all $C\in\mathcal{P}_{f}(\mathbb{N})$ we will have

[TABLE]

We prove (5) by induction on $k$. First observe that

[TABLE]

Also for $k+1$, recall the subsets $B_{i}$ in definition of the arithmetic progressions $P^{k+1}_{i}$, so we have

[TABLE]

where $D=\bigcup_{i\in C}B_{i}$. This proves (5). The proof now proceeds as in the proof of Theorem 3.1, in particular (1) can be proved easily for these new $P^{0}_{i}$. Now recall $P_{1}^{n}[1],P_{1}^{n}[2],\dots,P_{1}^{n}[n]$ so that for $s\in\{1,\dots,n\}$ and $C\in\mathcal{P}_{f}(\mathbb{N})$ we have

[TABLE]

Through induced coloring and this time using $=WB(l,c)$ we obtain $\gamma\in[c]$ and positive integers $a,d$ such that

[TABLE]

Again define the desire arithmetic progressions $Q_{i},i\in\mathbb{N}$ by

[TABLE]

Thus for all $C\in\mathcal{P}_{f}(\mathbb{N})$ we have

[TABLE]

Note that in the second and third equations from the end we have respectively used (5) and the easily checked fact $\displaystyle\sum_{i\in C}\operatorname{add}P_{i}^{n}=\operatorname{add}\displaystyle\bigoplus_{i\in C}\!P^{n}_{i}$. So we conclude that

[TABLE]

and the rest of the proof is the same as the proof of Theorem 3.1. ∎

4. Tower Bounds for the Finite Case

In this section we prove

Theorem 4.1**.**

For positive integers $n,c$ and $l\geq 3$, let $f(n,l,c)$ be the least positive integer $p$ such that whenever $\mathbf{c}$ is a $c$-coloring of $[p]$, then there are $l$-term arithmetic progressions $Q_{1},Q_{2},\dots,Q_{n}$ such that

• (i)

$Q_{1}\prec\cdots\prec Q_{n}$,

• (ii)

$\max(Q_{1}\oplus\cdots\oplus Q_{n})\leq p$,

• (iii)

there is $\gamma\in[c]$ such that for all $C\in\mathcal{P}^{+}([n])$ and all $s\in\{1,\dots,l\}$ we have

[TABLE]

Then $f(n,l,c)$ is a tower function.

Proof.

Let $q=W(l,c^{2^{\operatorname{Hind}(n,c)}})$, we will show that $f(n,l,c)\leq 2^{q^{3}}$. So from Gower’s elementary bounds for the van der Waerden numbers [4] and Theorem 2.4, it follows that $f(n,l,c)$ is a tower function. Suppose that $p\geq 2^{q^{3}}$ and $\mathbf{c}$ is a $c$-coloring of $[p]$. We show that $p$ satisfies the requirements of the theorem. Put $m=\operatorname{Hind}(n,c)$. Let $h_{i},1\leq i\leq m$ be positive integers defined by $h_{i}=(m+i)+(i-1)q$. For $1\leq i\leq m$, we define the $q$-term arithmetic progressions $P_{i}$ as follows

[TABLE]

Clearly $P_{1}\prec P_{2}\prec\cdots\prec P_{m}$. We claim that for each $1\leq s\leq q$, the positive integers $P_{1}[s],P_{2}[s],\dots,P_{m}[s]$ are pairwise power-disjoint. Let $1\leq s\leq q$, $2^{u}\leq q-1<2^{u+1}$ and $s-1=2^{u_{1}}+\dots+2^{u_{k}}$ with $u_{1}<u_{2}<\cdots<u_{k}$, hence $u_{k}\leq u\leq q-1$. Also from $i\leq m<h_{1}\leq h_{i}$ and

[TABLE]

it follows that

[TABLE]

for $1\leq i\leq m$. Now to prove the claim it would be enough to show that $A_{1},\dots,A_{m}$ are pairwise disjoint. In fact we show that

[TABLE]

which easily implies the disjointness of $A_{1},\dots,A_{m}$. First observe that

[TABLE]

Also for $1\leq i\leq m-1$ we have

[TABLE]

thus the claim is proved. Also we have

[TABLE]

Now we define a coloring $\mathbf{c}^{*}$ on $[q]$ as follows. For $u,v\in[q]$, we put $\mathbf{c}^{*}(u)=\mathbf{c}^{*}(v)$ if for all $B\in\mathcal{P}^{+}([m])$ we have

[TABLE]

Obviously the number of colors is $c^{2^{m}-1}$, so from $q=W(l,c^{2^{m}})$ it follows that there are $a,a+d,\dots,a+(l-1)d$ in $\{1,2,\dots,q\}$ such that

[TABLE]

which means that for all $B\in\mathcal{P}^{+}([m])$ and all $k_{1},k_{2}\in\{0,\dots,l-1\}$ we have

[TABLE]

We denote the above color by $\pi(B)$. So we have the well-defined function

[TABLE]

Now consider the following $m$-elements set of power-disjoint (due to the claim) positive integers

[TABLE]

From $m=\operatorname{Hind}(n,c)$ we infer that there exist $B_{1}<B_{2}<\cdots<B_{n}$ in $\mathcal{P}^{+}([m])$ and $\gamma\in[c]$ so that for all $C\in NU\{B_{1},\dots,B_{n}\}$ we have

[TABLE]

The desired arithmetic progressions $Q_{1},\dots,Q_{n}$ are defined as follows. For $1\leq i\leq n$, we set

[TABLE]

Obviously $Q_{1}\prec Q_{2}\prec\cdots\prec Q_{n}$ and from $B_{1}<B_{2}<\cdots<B_{n}$ it is easily seen that

[TABLE]

Now for $C\in\mathcal{P}^{+}([n])$ and $1\leq s\leq l$ we have

[TABLE]

where $D=\bigcup_{i\in C}B_{i}\in NU\{B_{1},\dots,B_{n}\}$. This finishes the proof of the theorem. ∎