Extremal sequences with forbidden sums (Q1363994)

Let \(s=(s_1,s_2,\dots,s_l)\) be a finite sequence of positive integers with length \(l=l(s)\), and let \(n(s)= s_1+\cdots+ s_l\). \(s'\) is said to be a fragment of \(s\) (using the notation \(s'\subseteq s\)) if for some \(i\), \(j\) \((1\leq i<j\leq l(s))\) \(s'=(s_i,s_{i+1},\dots, s_j)\). Let \(\text{Ex}(l,F)= \{s:l(s)=l\) and \(n(s')\not\in F\), \(\forall s'\subseteq s\}\), where \(F\) is a finite sequence of positive integers. A sequence \(s\) is called \(F\)-exclusive if it does not contain a fragment with a sum of \(F\). Let us denote by \(\text{Gex}(F)\) an infinite \(F\)-exclusive lexicographically minimal sequence and \(\text{Gex}(l,F)\) its initial fragment of length \(l\). Finally, let \(\text{gex}(l,F)= n(s)\), \(s=\text{Gex}(l,F)\) and \(\text{ex}(l,F)= \min\{n(s): s\in\text{Ex}(l,F)\}\). The author proves: the functions ex and gex are both uniform, i.e. \(\text{ex}(il,iF)=i\text{ex}(l,F)\) for all \(i\), \(l\), \(F\), and for every \(F\) the sequence \(\text{Gex}(F)\) is periodic from a definite starting point.