Some new additive and multiplicative Ramsey numbers

For $a,r\in{\mathbb N}$, the set of positive integers, define $FSP_2(a,r)$ (respectively $SP_2(a,r)$) to be the first $n\in{\mathbb N}$, if such exists, such that whenever $\{1,2,\ldots,n\}$ is $r$-colored, there exist $x$ and $y$ with $a\leq x<y$ such that $\{x,y,x+y,xy\}$ is monochromatic (respectively $\{x+y,xy\}$ is monochromatic). If no such $n$ exists, the number is defined to be infinite. It is an old result of R. Graham that $SP_2(a,2)$ is finite for all $a$. With that exception, the only cases (with $r>1$) for which $FSP_2(a,r)$ or $SP_2(a,r)$ are known to be finite are those for which explicit values have been computed. In this paper, we provide exact values of $FSP_2(a,2)$ for $a\leq 5$ (of which $FSP_2(1,2)$ and $FSP_2(2,2)$ were previously known). We provide exact values of $SP_2(a,3)$ for $a\leq 9$ and exact values of $SP_2(a,2)$ for $a\leq 105$. We also compute upper and lower bounds for $SP_2(a,2)$.

2010 Mathematics Subject Classification

05D10

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Published 16 May 2013