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We prove subadditivity of $\\chi_q(n)$ with respect to $n$, and then establish the following stronger recursive bound: \\[ \\chi_q(n)\\le \\chi_q(d)+\\chi_q(n+1-d)-1 \\] for all $1 \\leq d < n$. We use it to prove new upper bounds on $\\chi_q(n)$. For $q = 2$, using this recursion we prove that \\[ \\chi_2(n) \\le \\lfloor 2n/3 \\rfloor + 1 \\] for all $n \\ge 2$, and we show that this bound is tight for all $n \\le 7$. 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We prove subadditivity of $\\chi_q(n)$ with respect to $n$, and then establish the following stronger recursive bound: \\[ \\chi_q(n)\\le \\chi_q(d)+\\chi_q(n+1-d)-1 \\] for all $1 \\leq d < n$. We use it to prove new upper bounds on $\\chi_q(n)$. For $q = 2$, using this recursion we prove that \\[ \\chi_2(n) \\le \\lfloor 2n/3 \\rfloor + 1 \\] for all $n \\ge 2$, and we show that this bound is tight for all $n \\le 7$. 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