## 7 Citations

On improving a Schur-type theorem in shifted primes

- Mathematics
- 2021

We show that if N ≥ exp(exp(exp(k))), then any k-colouring of the primes that are less than N contains a monochromatic solution to p1 − p2 = p3 − 1.

Rado's criterion over squares and higher powers

- MathematicsJournal of the European Mathematical Society
- 2021

We establish partition regularity of the generalised Pythagorean equation in five or more variables. Furthermore, we show how Rado's characterisation of a partition regular equation remains valid…

Bootstrapping partition regularity of linear systems

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2020

Abstract Suppose that A is a k × d matrix of integers and write $\Re _A:{\mathbb N}\to {\mathbb N}\cup \{ \infty \} $ for the function taking r to the largest N such that there is an r-colouring…

## References

SHOWING 1-10 OF 10 REFERENCES

The Number of Monochromatic Schur Triples

- MathematicsEur. J. Comb.
- 1999

In this paper, we prove that in every 2-coloring of the set {1,? , N } =R?B, one can find at least N2/22 +O(N) monochromatic solutions of the equation x+y=z. This solves a problem of Graham et al. .

Roth's theorem in the primes

- Mathematics
- 2003

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the so-called Hardy-Littlewood…

A 2-Coloring of [1, N] Can Have (1/22)N2+O(N) Monochromatic Schur Triples, But Not less!

- MathematicsElectron. J. Comb.
- 1998

It is proved that the statement of the title, thereby solving the $100 problem of Ron Graham, is correct.

The primes contain arbitrarily long arithmetic progressions

- Mathematics
- 2004

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of…

On sets of integers containing k elements in arithmetic progression

- Mathematics
- 1975

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic…

On Λ(p)-subsets of squares

- Mathematics
- 1989

This paper is a follow up of [B1]. It is shown that the sequence of squares {n2|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic…

E-mail address: haopan79@yahoo.com.cn Department of Mathematics People's Republic of China

- E-mail address: haopan79@yahoo.com.cn Department of Mathematics People's Republic of China
- 2002

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

- Mathematics
- 1993

Beweis einer Baudet'schen Vermutung, Nieuw Arch

- Wisk
- 1927

Jahresb. Deutsche Math. Verein

- Jahresb. Deutsche Math. Verein
- 1916