For any homogeneous linear equation over F_p, every solution-free set whose Cayley graph has sublinear independence number has sublinear size exactly when some nonempty subset of the coefficients sums to zero.
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On a Ramsey--Tur\'{a}n variant of Roth's theorem
For any homogeneous linear equation over F_p, every solution-free set whose Cayley graph has sublinear independence number has sublinear size exactly when some nonempty subset of the coefficients sums to zero.