Exact values and lower bounds on the n-color weak Schur numbers for $$n=2,3$$

Abstract

AbstractFor integers k, n with k, $$n \geqslant 1$$ n ⩾ 1 , the n-color weak Schur number$$W\hspace{-0.6mm}S_{k}(n)$$ W S k ( n ) is defined as the least integer N, such that for every n-coloring of the integer interval [1, N], there exists a monochromatic solution $$x_{1},\dots , x_{k}, x_{k+1}$$ x 1 , ⋯ , x k , x k + 1 in that interval to the equation: $$\begin{aligned} x_{1}+x_{2}+\dots +x_{k} =x_{k+1}, \end{aligned}$$ x 1 + x 2 + ⋯ + x k = x k + 1 , with $$x_{i} \ne x_{j}$$ x i ≠ x j , when $$i\ne j.$$ i ≠ j . In this paper, we obtain the exact values of $$WS_{6}(2)=166$$ W S 6 ( 2 ) = 166 ,       $$WS_{7}(2)=253$$ W S 7 ( 2 ) = 253 , $$WS_{3}(3)=94$$ W S 3 ( 3 ) = 94 and $$WS_{4}(3)=259$$ W S 4 ( 3 ) = 259 and we show new lower bounds on n-color weak Schur number $$W\hspace{-0.6mm}S_{k}(n)$$ W S k ( n ) for $$n=2,3.$$ n = 2 , 3 .