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Assume that for some α < 1 and for all nutural n a set Fn of at most 2 αn "forbidden" binary strings of length n is fixed. Then there exists an infinite binary sequence ω that does not have (long) forbidden substrings. We prove this combinatorial statement by translating it into a statement about Kolmogorov complexity and compare this proof with a combinatorial one based on Laslo Lovasz local lemma. Then we construct an almost periodic sequence with the same property (thus combines the results from [1] and [2]). Both the combinatorial proof and Kolmogorov complexity argument can be generalized to the multidimensional case.

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