Monochromatic Hilbert cubes and arithmetic progressions

The Van der Waerden number $W(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic arithmetic progression of length $k$. Similarly, the Hilbert cube number $h(k,r)$ denotes the smallest $n$ such that whenever $[n]$ is $r$--colored there exists a monochromatic affine $k$--cube, that is, a set of the form$$\left\{x_0 + \sum_{b \in B} b : B \subseteq A\right\}$$ for some $|A|=k$ and $x_0 \in \mathbb{Z}$. We show the following relation between the Hilbert cube number and the Van der Waerden number. Let $k \geq 3$ be an integer. Then for every $\epsilon>0$, there is a $c>0$ such that $$h(k,4) \ge \min\{W(\lfloor c k^2\rfloor, 2), 2^{k^{2.5-\epsilon}}\}.$$ Thus we improve upon state of the art lower bounds for $h(k,4)$ conditional on $W(k,2)$ being significantly larger than $2^k$. In the other direction, this shows that the if the Hilbert cube number is close its state of the art lower bounds, then $W(k,2)$ is at most doubly exponential in $k$. We also show the optimal result that for any Sidon set $A \subset \mathbb{Z}$, one has $$\left|\left\{\sum_{b \in B} b : B \subseteq A\right\}\right| = \Omega( |A|^3) .$$


Introduction
A k-term arithmetic progression (AP) in the integers is a set of the form where x 0 , d ∈ Z. Recall the famous Van der Waerden theorem. Theorem 1.1 (Van der Waerden [Wa], 1927). Let k, r ≥ 2 be integers. Then there exists an n such that in any r-coloring of [n], at least one color class contains a k-term AP.
The smallest such n is said to be the Van der Waerden number, which we denote by W (k, r). The state of the art bounds on W (k, r) are as follows: Berlekamp [Be] showed for prime p we have p · 2 p ≤ W (p + 1, 2). This result was recently generalized by Blankenship, Cummings and Taranchuk [BCT] who showed the following for p prime p r−1 2 p ≤ W (p + 1, r).
(1) Kozik and Shabanov [KoSh] proved the general lower bound c · r k−1 ≤ W (k, r) for all k ≥ 3, which is a slight improvement over an application of the Lovász local lemma [Sza]. The best known upper bound for W (k, r) is the breakthrough result of Gowers [Go] Theorem 1.2 (Hilbert [Hi], 1892). Let k, r ≥ 2 be integers. Then there exists an n such that any r-coloring of [n], at least one color class contains an affine k-cube.
Since every k 2 -term AP is an affine k-cube, we have h(k, r) ≤ W k 2 , r .
Thus Van der Waerden's theorem implies Theorem 1.2 (but not the bound in (3)). Szemerédi [Sze], in his seminal paper on the density version of Van der Waerden's theorem, proved that h(k, r) = O(r) 2 k . Hilbert's and Szemerédi's results are a massive improvement over combining (4) with the state of the art Van der Waerden bounds in (1). The case where k = 2 was asymptotically solved by Brown, Chung, Erdős and Graham [BCEG], who showed that h(2, r) = (1 + o(1))r 2 .
Their lower bound uses difference sets arising from finite projective planes, and their upper bound follows from bounds on Sidon sets. Gunderson and Rödl [GuRö] showed that for k ≥ 3 we have Conlon, Fox and Sudakov [CFS] improved the bound of Erdős and Spencer [ErSp] by showing that there exists an absolute constant c such that This is currently the best lower bound known for small values of r. Their proof heavily relies on an inverse Littlewood-Offord type theorem of Nguyen and Vu [NgVu], which we will also use in our proof of our main result. Note that a significant improvement on (5) would improve on (1) because of (4). Unfortunately, improving the bounds on Van der Waerden numbers is a notoriously difficult problem. To circumvent this problem, in this paper we focus on improving the Conlon-Fox-Sudakov bound conditional on the fact that W (k, 2) is much bigger than 2 k . Theorem 1.3. Let k ≥ 3 be an integer. Then for every ǫ > 0, there is a c > 0 such that h(k, 4) ≥ min{W (⌊ck 2 ⌋, 2), 2 k 2.5−ǫ }.
Theorem 1.3 asserts that either (i) the lower bound for W (k, 2) in (1) is far from sharp and h(k, 4) is larger than (5), (ii) the lower bound for W (k, 2) in (1) is nearly sharp and we can roughly reverse (4).
We remark that by Theorem 1.3, one can solve Graham's conjecture in (2) by providing an upper bound of h(k, 4) < 2 k 2.5−ǫ . Our proof of Theorem 1.3 can be easily adapted to provide lower bounds for h(k, r) where r > 1 is a square of an integer. We briefly mention that Hilbert cubes have played a central role in upper bounds for van der Waerden numbers [Go, Sze], via Gower's uniformity norms and Szemerédi's cube lemma.
The idea for the proof of Theorem 1.3 is the following. In a random coloring of [n], the probability that an affine k-cube, x 0 + Σ * A is monochromatic is This probability is small when |Σ * A| is large. When |Σ * A| is small, then A should look much like an AP and we are led back to the Van der Waerden problem. Our main tools for making this argument rigorous are a paper of Nguyen and Vu [NgVu] (see Theorem 3.1) concerning the Littlewood-Offord theory and another paper of Szemerédi and Vu [SzVu] (see Theorem 3.2) on finding long APs in restricted sumsets, along with some analysis of our own (see Lemma 4.1) of the case when A is a large subset of a generalized AP.
To prove Theorem 1.3, we analyze which A ⊂ Z satisfy We conclude this implies A has some additive structure, which eventually yields Theorem 1.3. Curiously, after this analysis we cannot rule out the case that A is a Sidon set, that is |A + A| = |A|+1 2 . We use different techniques to handle this case.
Theorem 1.4. There exists a c > 0 such that for any Sidon set A ⊂ Z one has This result is a side product of our methods and we believe it is of independent interest. The proof is elementary, self-contained and best possible up to the constant c. We briefly mention a related theorem of finding monochromatic Folkman cubes, a wide generalization of Schur's theorem that was obtained independently by Folkman, Rado and Sanders. This generalization is now commonly referred to as Folkman's theorem (see for example [GRS]). Let F (k, r) be the smallest n such in that any r-coloring of [n] one can find a set A of size k such that Σ * A ⊆ [n] and Σ * A is monochromatic. The state of the art bounds on F (k, r) are significantly different from the best bounds on H(k, r). Indeed, already for F (k, 2) the best upper bound due to Taylor [Ta] is tower-type, while the best lower bound is due to Balogh-Eberhard-Narayanan-Treglown-Wagner [BENTW]. They are as follows: where the tower on the right side has height 4k − 3.

Initial set-up and the random coloring
Generalized APs play a central role in our argument.
Definition 2.1. A generalized AP (GAP) of rank r is a set of the form for some a, m 1 , . . . , m r , M 1 , . . . , M r ∈ Z, and d 1 , Proof idea. We let N = min{W (⌊ck 2 ⌋, 2), 2 k 2.5−ǫ /(10 log k) } − 1 where c > 0 is a sufficiently small, fixed constant that depends on our argument. We color [N ] by a product coloring is a uniformly random coloring.
If a Hilbert cube has many distinct elements, then the coloring χ 2 makes sure it is not monochromatic. We will show that all Hilbert cubes having very few distinct elements will contain a ck 2 -term AP, in which case χ 1 ensures it is not monochromatic.
To understand χ 2 , we will need the following lemma, which appears in a short paper of Erdős and Spencer [ErSp].
Lemma 2.1. Let n, k, u ∈ N be integers with u ≥ k(k + 1)/2. The number of sets S ⊆ [n] of size k satisfying |Σ * S| ≤ u is at most (kn) log u u 2k .
The critical case for our purposes is u = k a for some a = O(1) and k a fixed power of log n and so the bound in Lemma 2.1 is n a log k(1+o (1)) . In this case it is easy to see that ∼ k n a proper GAPs of rank a − 1 satisfy the hypothesis of Lemma 2.1. The additional log k in the exponent is not concerning for our purposes. A corollary of Lemma 2.1 is that a random coloring is unlikely to contain Hilbert cubes of large size.
Corollary 2.2. Fix an arbitrary a > 2. If and χ 2 : [N ] → {0, 1} is the uniform random 2-coloring then w.h.p. (as k → ∞, a fixed) χ 2 does not contain a monochromatic Hilbert cube of size at least k a .
Proof. The probability that a Hilbert cube of size u is monochromatic under χ 2 is 2 1−u . By Lemma 2.1 the number of such cubes is ≤ N (kN ) log u u 2k , since we have at most N choices for x 0 . By the union bound and (6) the probability, p(k), that there is a monochromatic Hilbert cube of size at least k a satisfies (7) 3. The AP-avoiding coloring To analyze a Hilbert cube x 0 + Σ * A, we ignore x 0 and focus on the structure of A. We assume that since we Corollary 2.2 implies we may choose a χ 2 so that all Hilbert cubes not satisfying (8) are not monochromatic.
We proceed in several steps. First, we use a result of Nguyen and Vu [NgVu] to show that for sets satisfying (8), at least half of the set A must be contained in a GAP of small rank and volume. We consider two cases for the rank of the resulting GAP, and show that in each case 3.1. Results concerning restricted sumsets of GAPs. We first recall an inverse theorem of Nguyen and Vu.
Theorem 3.1 (Nguyen-Vu, special case of Theorem 2.1 in [NgVu]). Let C be a constant, and let A be a k-element set with |Σ * A| ≤ k C . Then there is a proper symmetric rank r GAP, Q, such that |A ∩ Q| ≥ 1 2 k and |Q| = O(k C−r/2 ), where the constant factor may depend on C. In particular, if A satisfies (8), then the two possible cases are (1) |Q| = O(k 2−ǫ ) and Q is a rank 1 GAP (an AP), (2) |Q| = O(k 1.5−ǫ ) and Q is a rank 2 GAP.
To see the last point of Theorem 3.1, note that if Q were to have rank 3 or greater, then |Q| = O(k 1−ǫ ), which contradicts that Q contains at least half of the elements of A.
Definition 3.1. For an integer ℓ, let ℓ * A denote the subset of Σ * A consisting of sums of exactly ℓ distinct elements Theorem 3.2 (Szemerédi-Vu, Theorem 7.1 in [SzVu]). For any fixed positive integer r there are positive constants C and c depending on r such that the following holds. For any positive integers n and ℓ and any set A ⊆ [n] with ℓ ≤ 1 2 |A| and ℓ r |A| ≥ Cn, the set ℓ * A contains a proper GAP of rank r ′ and size at least cℓ r ′ |A|, for some integer r ′ ≤ r.
Our standing assumption (8) is not compatible with r ′ ≥ 2 and ℓ = Ω(|A|) in Theorem 3.2 as long as n = |A| O(1) . We formulate this in the following corollary.
Corollary 3.3. For every positive integer r, there is a constant C ′ such that the following holds. Let P be an arbitrary AP with |P | = n, and let A ⊆ P which satisfies C ′ n ≤ |A| r+1 and |Σ * (A)| = o(|A| 3 ). Then Σ * A contains an AP of length Ω r (|A| 2 ).
Proof. If P = {a + kd : 1 ≤ k ≤ n}, then apply the Freiman homomorphism x → x−a d maps P to [n], A to a subset of [n], and preserves the size of both A and Σ * A. So we may assume that P = [n] in what follows.
The following corollary is an immediate consequence of Corollary 3.3 and our choice of χ 1 .
Corollary 3.4. Suppose a set A is of size k and contained in an AP of size O(k α ) for some α ≥ 1. Then x 0 + Σ * A is not monochromatic in χ 1 .
In case (1) of Theorem 3.1, we have that A is a subset of an AP of length O(k 2−ǫ ), and so by Corollary 3.4, x 0 + Σ * A is not monochromatic in the coloring χ 1 . Thus x 0 + Σ * A is not monochromatic in the product coloring χ 1 × χ 2 .

3.2.
Completing the proof of Theorem 1.3. We are now left to analyze case (2) in Theorem 3.1. Here at least half of the elements of A is contained in a proper, symmetric GAP, Q, of rank 2 and size O(k 1.5−ǫ ). In this case, A is basically a dense subset of a two-dimensional integer box (ignoring the technicality that while Q is proper, it may not be that |Σ * Q| = 2 |Q| ). In this case, the size of Σ * A is roughly cubic in |A| as is shown by the following lemma.

Lemma 3.5. There is an absolute constant C such that for
This is best possible up to the logarithm, as is seen by taking A = [m]×[n]. We prove Lemma 3.5 in the following section, but first show how it implies Theorem 1.3.
Since Q is proper, we may decompose it into the following six disjoint sets: At least one of |A ∩ Q 1 |, |A ∩ Q 2 |, . . . , |A ∩ Q 6 | has size at least 1 12 |A|. If this happens for Q 5 or Q 6 , then we are in the situation of case (1) of Theorem 3.1, which we already handled in Corollary 3.4.
Without loss of generality, by switching the signs of d 1 and d 2 , we may assume |A ∩ Q 1 | ≥ |A| 12 . Let φ : Q 1 → Z 2 via φ(id 1 + jd 2 ) = (i, j). Since Q 1 is proper, φ is injective. It follows, for k sufficiently large, that we have Thus we may apply Lemma 3.5 to find that Combing this with (8), we have that One may compare the rest of our argument to [TV,Theorem 3.40]. It follows that there is a "collision," that is there exist 1 ≤ x 1 , y 1 ≤ km and 1 ≤ x 2 , y 2 ≤ kn, satisfying This simplifies to |d 1 | · |x 1 − y 1 | = |d 2 | · |x 2 − y 2 |.
Similarly, |d 1 | d ≤ km. Let R be the AP with common difference d given below: Then R contains every integer divisible by d between the largest and smallest element of Q, so Q ⊆ R. Moreover, By Corollary 3.4, x 0 + Σ * (A ∩ Q 1 ) is not monochromatic in χ 1 and so neither is x 0 + Σ * A. Thus we have handled case (2) of Theorem 3.1 and completed the proof of Theorem 1.3.

Restricted sumsets for dense subsets of high dimensional boxes
It remains to prove Lemma 3.5. We work in arbitrary dimensions, which may be of independent interest. In the following lemma, we are only interested in the case m = 1, but other values of m are useful as a strengthened induction hypothesis. We recall the m-fold sum is where there are m summands.

Suppose that
From the estimate |X + Y | ≥ |X| + |Y | − 1 we have |mX| ≥ m|X| − m + 1, and therefore as long as |A| ≥ 2 we have |mΣ * A| ≥ m |A|+1 2 − m + 1 ≥ 1 2 |A| 2 m. For the induction step, assume that this lemma holds in dimension Then the total number of elements of A contained in sparse stacks is at most so at least 1 2 |A| elements of A are in dense stacks. The sizes of dense stacks range from 1 so that X t indices all stacks whose sizes range from t to 2t. By a dyadic decomposition, we can find a t so that the union of the stacks indexed by X t is large. That is, letting s = log 2 α −1 , we can partition the indices of all the dense stacks into the disjoint union of s sets Since at least 1 2 |A| elements of A are in dense stacks, there must be a t = 2 i−1 αN d for some i between 0 and s − 1 such that at least 1 2s |A| elements of A are in stacks indexed by some x ∈ X t . For each x ∈ X t , we choose two disjoint sets B We will show that |mΣ * A| is large in two steps.
Let b ∈ mΣ * A be given by summing the ⌊ t 3 ⌋ smallest elements of each B x , each with multiplicity m. Then b + mΣ * C is a subset of mΣ * A. We show that not only is |b + mΣ * C| large, but that its projection onto the first d − 1 coordinates is large.
In this projection, the exact elements of each C x are irrelevant, since their first d − 1 coordinates are just x. Being able to choose the elements of C x , of which there are at least t 4 , up to m times each is equivalent to being able to include x in a sum up to mt 4 times, and so the size of the projection is | mt 4 Σ * X t |. Since each stack A x has size at most 2t, and the union of all stacks indexed by X t has size at least |A| 2s , we know that X t itself must have size at least |A| 4st . We apply the induction hypothesis to X t . The density of X t in [N 1 ] × · · · × [N d−1 ] is at least which will satisfy the conditions in the induction hypothesis provided we choose C d sufficiently large compared to C d−1 . Additionally, log α ′−1 = Θ(log α −1 ). So we have Second, for each element of b + mΣ * C, we show that there are many elements of mΣ * A with the same projection onto the first d − 1 coordinates. We can obtain such elements by replacing b with a different sum which also uses m⌊ t 3 ⌋ elements of B x for each x, counting multiplicity. Let k = ⌊ t 3 ⌋, and let B x = {b x,1 , . . . , b x,2k }. For a fixed x, there are at least mk 2 distinct sums of elements of B x with total multiplicity mk. The argument here is similar to the d = 1 case of this lemma. Start with the smallest possible sum, For each sum with total multiplicity mk, we may increase its d th coordinate by choosing the largest i < 2k such that b x,i is included in the sum, while b x,i+1 is included fewer than m times, and replace b x,i by b x,i+1 . This ends only when we reach the largest possible sum, The sum of the indices on the mb x,i , taken with multiplicity, starts at k i=1 mi = m k+1 2 and ends at 2k i=k+1 mi = mk 2 + m k+1 2 . In each step, since we replace some b x,i by b x,i+1 , the sum of indices increase by 1, so we take a total of mk 2 ≥ m( t 4 ) 2 = Ω(mt 2 ) steps. Now we must aggregate this result over all x ∈ X t . Starting at the element b as previously defined, go through the elements of X t arbitrarily, and for each x ∈ X t , perform the above process, taking Ω(mt 2 ) steps. There are a total of |X t | · Ω(mt 2 ) steps taken, and each one increases the d th coordinate while leaving the first d − 1 coordinates unchanged. Altogether, for every a ∈ b + mΣ * C, we obtain |X t | · Ω(mt 2 ) = Ω mt 2 · |A| 4st = Ω |A|mt log α −1 different elements of mΣ * A with the same first d − 1 coordinates as a.
Repeating this for each of the Ω |A| d m d−1 t −1 log 10d−10+d |A| elements of b + mΣ * C, we get In our application we will have d = 2. In this case, assuming that A is sufficiently large, we get the bound in a second lemma, given below. By applying a Freiman isomorphism, that bound also applies when d > 2 but the set A is too sparse to use Lemma 4.1 directly.
Lemma 4.2. There is an absolute constant C such that for all d ≥ 2, the following holds. Then Proof. First we handle the case d = 2. Then this result is a direct application of Lemma 4.1, once we assure ourselves that it applies. Without loss of generality, assume that N 1 ≤ N 2 .
Since |A| ≥ C √ N 1 N 2 , the density α = |A| N 1 N 2 satisfies αN 2 ≥ C |N 2 | |N 1 | ≥ C, so the conditions of Lemma 4.1 are satisfied for any C which is at least the constant C 2 from that lemma.
Taking m = 1, we conclude that Next, we assume that d > 2 and we begin by omitting any coordinates i with N i = 1, so that we may assume N i ≥ 2 for all i. Therefore if we define The homomorphism φ is injective on [N 1 ] × · · · × [N d ], so the image φ(A) has the same size as A. Therefore which is large enough for the d = 2 case of this lemma to apply if we choose C = C 2 √ 2. Applying the d = 2 case of this lemma, |Σ * A| ≥ |Σ * φ(A)| ≥ Ω |φ(A)| 3 (log |φ(A)|) 4 = Ω |A| 3 (log |A|) 4 , which was what we wanted.
Lemma 3.5 follows from Lemma 4.2 by taking d = 2.

Sidon sets
We now set out to prove Theorem 1.4. We prove this by starting with a small subset X ⊆ A, and adding elements to X slowly while ensuring that |Σ * X| grows quickly. In the end, |Σ * X| will reach Ω(|A| 3 ) in size before the set A is exhausted.
As long as |Σ * X| is relatively small, the following lemma guarantees that we can increase |Σ * X| by a factor of 3 2 with the addition of only two new elements.
Proof. Let B = {a 1 + a 2 : a 1 , a 2 ∈ A \ X}. Since A is Sidon, all elements of B are distinct, so |B| = |A\X| 2 ≥ 1 2 |A| 2 . In particular, |B| ≥ |Σ * X|. The total number of solutions of the equation s 1 + b = s 2 with s 1 , s 2 ∈ Σ * X and b ∈ B is at most |Σ * X| 2 : once we choose the set {s 1 , s 2 }, we are forced to choose the order s 1 < s 2 , and