AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions to x+y<z over all 2-colorings of {k+1,k+2,…,k+n}. We show that for any k⩾0, Mk(n)=Cn3(1+ok(1)), where C=112(1+22)2≈0.005686. A structural result is also proven, which can be used to determine the exact value of Mk(n) for given k and n
AbstractFor positive integers a1,a2,…,am, we determine the least positive integer R(a1,…,am) such th...
AbstractLet Kn,n be the complete bipartite graph with n vertices in each partition. We denote M(C4,K...
AbstractLet V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-colo...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
AbstractIn this paper, we prove that in every 2-coloring of the set {1,⋯ , N } =R∪B, one can find at...
For k∈N, write S(k) for the largest natural number such that there is a k-colouring of {1,…,S(k)} wi...
AbstractThere exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromati...
Suppose that $\mathbb{N}$ is 2-coloured. Then there are infinitely many monochromatic solutions to $...
AbstractFor all integers m⩾3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent t...
Let G be a finite cyclic group an let r be a positive integer. Let A be a k×m matrix with integer en...
AbstractFor integers c⩾0 and k⩾1, let R=R(c,k) be the least integer, provided it exists, such that e...
For relatively prime positive integers a and b, let n = R(a,b) denote the least positive integer suc...
AbstractIf F and G are graphs, define M(G,F) to be the minimum number of monochromatic G that occur ...
Given an equation, the integers [n] = {1,2,...,n} as inputs, and the colors red and blue, how can we...
We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-col...
AbstractFor positive integers a1,a2,…,am, we determine the least positive integer R(a1,…,am) such th...
AbstractLet Kn,n be the complete bipartite graph with n vertices in each partition. We denote M(C4,K...
AbstractLet V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-colo...
AbstractFor integers n⩾1 and k⩾0, let Mk(n) represent the minimum number of monochromatic solutions ...
AbstractIn this paper, we prove that in every 2-coloring of the set {1,⋯ , N } =R∪B, one can find at...
For k∈N, write S(k) for the largest natural number such that there is a k-colouring of {1,…,S(k)} wi...
AbstractThere exists a minimum integer N such that any 2-coloring of {1,2,…,N} admits a monochromati...
Suppose that $\mathbb{N}$ is 2-coloured. Then there are infinitely many monochromatic solutions to $...
AbstractFor all integers m⩾3 and all natural numbers a1,a2,…,am−1, let n=R(a1,a2,…,am−1) represent t...
Let G be a finite cyclic group an let r be a positive integer. Let A be a k×m matrix with integer en...
AbstractFor integers c⩾0 and k⩾1, let R=R(c,k) be the least integer, provided it exists, such that e...
For relatively prime positive integers a and b, let n = R(a,b) denote the least positive integer suc...
AbstractIf F and G are graphs, define M(G,F) to be the minimum number of monochromatic G that occur ...
Given an equation, the integers [n] = {1,2,...,n} as inputs, and the colors red and blue, how can we...
We show that the number of monochromatic solutions of the equation x 1 α1x2 α2⋯x r αr = g in a 2-col...
AbstractFor positive integers a1,a2,…,am, we determine the least positive integer R(a1,…,am) such th...
AbstractLet Kn,n be the complete bipartite graph with n vertices in each partition. We denote M(C4,K...
AbstractLet V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-colo...
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