# Decomposability in a context of size constraints on intervals

Let $F$ be a finite set of pairs of positive integers. Say that a set $A \subseteq {\mathbb Z}$ is $F$-admissible iff its intersection with any integer interval of length $a$ has cardinality at most $b$, for any $(a,b) \in F$.

Note that the notion of admissiblity does not change if we add the pair $(1,1)$ to $F$, because an intersection with a point has certainly cardinality at most $1$. So we assume that $(1,1) \in F$.

For any $n\geq 1$, denote by $\mu_F(n)$ the largest cardinality of an $F$-admissible subset of $\lbrace 1,2, \ldots , n \rbrace$. I say that $n$ is decomposable if we can find a sequence $(a_1,b_1), \ldots ,(a_r,b_r)$ of pairs in $F$, such that

$$n=\sum_{k=1}^r a_k, \ \mu_F(n)=\sum_{k=1}^r b_k$$

Is it true that any $n\geq 1$ is decomposable ?

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