Finding linear patterns of complexity one

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s≥2, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x₁,⋯,x_s as well as the averages (x_i+x_j)/2 (1≤i<j≤s). Our main result states that if a set A∩[N] has density δ > (log N)^-c(s) for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. This improves on the previous bound of the form δ > (log log N)^-c(s) due to Dousse [3]. We also deduce, as a corollary, an improvement of a problem involving sum-free subsets.