The most straightforward, but quick way to do this, is via a backtracking algorithm. Using the colours $0$ and $1\text{,}$ assume we have a colouring of $[1,n-1]$ that does not contain a monochromatic set of the form $a,b,a+b,ab\text{.}$ We then attempt to colour integer $n\text{.}$ We start by assigning integer $n$ colour $0\text{.}$ We then only need to check if our colouring of $[1,n]$ avoids monochromatic sets of the form $a,b,a+b,ab$ where one of $a+b$ or $ab$ equals $n\text{.}$ If no monochromatic set, we move on to colouring $[1,n+1]\text{.}$ If there is a monochromatic set, we change the colour of integer $n$ to $1$ and check again. If integer $n$ being either colour $0$ and $1$ results in a monochromatic set, then we backtrack to integer $n-1\text{.}$ If it is colour $0$ then we change it to colour 1 and repeat the monochromatic check. If it is colour 1, then we backtrack again to $n-2\text{.}$ This will eventually halt provided we are guaranteed a monochromatic set (which we are with 2 colours).
By the what is stated in the problem, for the case where $a\not= b\text{,}$ the returned number will be $252\text{,}$ with the case $a = b\text{,}$ it is expected to return a number less than 252 which we do not know the exact value of.