Monochromatic Hilbert Cubes and Arithmetic Progressions

@article{Balogh2019MonochromaticHC,
  title={Monochromatic Hilbert Cubes and Arithmetic Progressions},
  author={J{\'o}zsef Balogh and Mikhail Lavrov and George Shakan and Adam Zsolt Wagner},
  journal={Electron. J. Comb.},
  year={2019},
  volume={26},
  pages={P2.22}
}
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… Expand

References

SHOWING 1-10 OF 21 REFERENCES
A quantitative improvement for Roth's theorem on arithmetic progressions
  • T. Bloom
  • Mathematics, Computer Science
  • J. Lond. Math. Soc.
  • 2016
We improve the quantitative estimate for Roth's theorem on three-term arithmetic progressions, showing that if $A\subset\{1,\ldots,N\}$ contains no non-trivial three-term arithmetic progressions thenExpand
Monochromatic sumsets
TLDR
A lower bound for F(k) is given, which states that at most (kn)'g"u lk k-sets Sc [ n ] have IP(S)I <u and there are at most lg u doubling i and at most n'g" choices for the values a i. Expand
Long arithmetic progressions in sumsets: Thresholds and bounds
l∗A = {a1 + · · ·+ al|ai ∈ Ai, ai = aj}. Among the most well-known results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently largeExpand
Extremal Problems for Affine Cubes of Integers
TLDR
This paper addresses both density and Ramsey-type questions for affine d-cubes, and improves for upper and lower bounds on h(d,r) are given for d>2. Expand
Ramsey Theory
Party Problem: Find the minimum number R(k, l) of guests that must be invited so that at least k will know each other or at least l will not know each other (we assume that if A knows B, then B knowsExpand
Improved algorithms for colorings of simple hypergraphs and applications
The paper deals with extremal problems concerning colorings of hypergraphs. By using a random recoloring algorithm we show that any n-uniform simple (i.e. every two distinct edges share at most oneExpand
A new lower bound for van der Waerden numbers
TLDR
This recurrence gives the lower bound w ( r, p + 1) > p r − 1 2 p when r ≤ p, which generalizes Berlekamp's theorem on 2-colorings, and gives the best known bound for a large interval of r. Expand
Quantitative Forms of a Theorem of Hilbert
TLDR
Hilbert needed this lemma in connection with certain results on the irreducibility of rational functions and never pursued the combinatorial directions to which it pointed, but others did, and in 1974, Hindman proved the much stronger result that in any partition of all the positive integers into finitely many classes, some class must contain an infinite projective cube. Expand
An improved lower bound for Folkman's theorem
Folkman's theorem asserts that for each k∈N, there exists a natural number n=F(k) such that whenever the elements of [n] are two-coloured, there exists a set A⊂[n] of size k with the property thatExpand
Optimal Inverse Littlewood-Offord theorems
Abstract Let η i , i = 1 , … , n , be iid Bernoulli random variables, taking values ±1 with probability 1 2 . Given a multiset V of n integers v 1 , … , v n , we define the concentration probabilityExpand
...
1
2
3
...