A subset of $[1, \dots, n]$ is said to be 3-AP (arithmetic progression) free if for all $x_1, x_2, x_3$ we have $x_1 + x_2 \ne 2x_3$.
I am interested in a generalization of this concept: subsets of $[1, \dots, n]$ with the property that for all $x_1, \dots, x_k$ we have $x_1 + \dots + x_k \ne kx_{k+1}$.
Is there a name for this in the literature? Have basic questions about it been answered -- for example, what is the largest "K-AP free" subset (terrible name, I know) of $[1, \dots, n]$?