Linear-dimension strengthening of the geometric Erdős–Hajnal conjectures

Use the notions of $c_0$-free colouring, $(c,s)$-homogeneous subspace, and induced restriction from the preceding conjectures.

Linear-dimension strengthening. The multicoloured geometric Erdős–Hajnal conjecture and the geometric Erdős–Hajnal conjecture for induced restrictions should both hold with $\delta=1$.

This would improve the guaranteed polynomial dimension to a linear bound in the ambient dimension. The source presents it as a strengthening suggested by Jacob Fox and records that no construction is known to refute it.