Large sets avoiding linear patterns

We prove that for any dimension function h h with h ≺ x d h \prec x^d and for any countable set of linear patterns, there exists a compact set E E with H h ( E ) > 0 \mathcal {H}^h(E)>0 avoiding all the given patterns. We also give several applications and recover results of Keleti, Maga, and Máthé.