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I am looking for any (finite) upper bound of the ramsey number $r(a_1,\ldots, a_m)$. I can prove the well known fact for any positive integers $a,b$ there is a $c$ for which $c\ge r(a,b)$ by taking $c=r(a-1,b)+r(a,b-1)$. Is this argument and its proof restricted to 2-tuples or would $r(a_1-1,a_2,\ldots ,a_m)+\ldots +r(a_1,\ldots,a_m-1)$ be an upper bound for $r(a_1,\ldots, a_m)$? My work shown (using induction) is the same steps required in the proof for the case $m=2$ but I am wondering if I have overlooked something.

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This survey paper gives the inequality (see equation $6.1(a)$) $$R(k_1, \ldots, k_r) \leq 2 - r + \sum_{i=1}^r R(k_1, \ldots, k_{i-1}, k_i - 1, k_{i+1}, \ldots, k_r).$$

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