DOIAbstract
AbstractIn 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.
AbstractIn 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.
AbstractSuppose that all primes are colored with k colors. Then there exist monochromatic primes p1,...
A Pythagorean triple is a triple of positive integers a, b, c ∈ N⁺ satisfying a² + b² = c². Is it tr...
We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-col...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
Suppose that $\mathbb{N}$ is 2-coloured. Then there are infinitely many monochromatic solutions to $...
Let L(t) represent the equation x1 + x2 + · · ·+ xt−1 = xt. For k> 1, 0 6 i 6 k − 1, and ti> ...
AbstractA classic result of I. Schur [9] asserts that for everyr⩾2 and fornsufficiently large, if th...
AbstractLet V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-colo...
Let a and r be positive integers. By definition, sar is the least positive integer such that, for an...
For k∈N, write S(k) for the largest natural number such that there is a k-colouring of {1,…,S(k)} wi...
A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a m...
We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular,...
AbstractThere exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromati...
AbstractLet S={1,2,…,n}, and let S=S1∪S2 be a partition of S in two disjoint subsets. A triple (a,b,...
AbstractSuppose that all primes are colored with k colors. Then there exist monochromatic primes p1,...
A Pythagorean triple is a triple of positive integers a, b, c ∈ N⁺ satisfying a² + b² = c². Is it tr...
We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-col...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
Suppose that $\mathbb{N}$ is 2-coloured. Then there are infinitely many monochromatic solutions to $...
Let L(t) represent the equation x1 + x2 + · · ·+ xt−1 = xt. For k> 1, 0 6 i 6 k − 1, and ti> ...
AbstractA classic result of I. Schur [9] asserts that for everyr⩾2 and fornsufficiently large, if th...
AbstractLet V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-colo...
Let a and r be positive integers. By definition, sar is the least positive integer such that, for an...
For k∈N, write S(k) for the largest natural number such that there is a k-colouring of {1,…,S(k)} wi...
A standard proof of Schur's Theorem yields that any $r$-coloring of $\{1,2,\dots,R_r-1\}$ yields a m...
We resolve the Ramsey problem for {x,y,z: x + y = p(z)} for all polynomials p over ℤ. In particular,...
AbstractThere exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromati...
AbstractLet S={1,2,…,n}, and let S=S1∪S2 be a partition of S in two disjoint subsets. A triple (a,b,...
AbstractSuppose that all primes are colored with k colors. Then there exist monochromatic primes p1,...
A Pythagorean triple is a triple of positive integers a, b, c ∈ N⁺ satisfying a² + b² = c². Is it tr...
We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-col...
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