Color any $67$ numbers in the set $M = \{1,2,\dots, 100 \}$ $\color{red}{\rm red}$ and the rest $\color{blue}{\rm blue}$.
If there exists a subset $A_{i,k} = \{i, i +1, \dots, i + 3k -1\}$ of $M$, with exactly $2k$ numbers in red and $k$ numbers in blue, find the maximum value of $k$ (meaning that such a subset exists regardless of how $M$ is colored).
By now, I claim that $k \not \ge 18$.
reasons were as follows
We could color $1\sim17$ and $85 \sim 100$ $\color{blue}{\rm blue}$, and the rest $\color{red}{\rm red}$.Then there don't exists a legal subset $A$ if $k\ge 18$.
