Let $a,b,c$ be positive integers with gcd$(a,b,c)=1$, and let $\mathbb{N}$ denote the set of nonnegative integers.

It is well known that $\mathbb{N} \setminus (a \mathbb{N}+b \mathbb{N} + c \mathbb{N})$ is a finite set, but it seems difficult to determine a formula for its maximum (the Frobenius number). This is the famous 'coin problem'.

I am asking about a related problem. Consider now the set of *vectors* $$(1,1,1) \mathbb{N} \setminus \sum (a,b,c) \mathbb{N},$$
where the sum ranges over all six permutations of $(a,b,c)$. Is it necessarily the case that, for relatively prime $a,b,c$, there is a maximum constant integral vector in this set? If so, what can we say about the maximum?

I suspet this may be connected (via a clever argument) with the ordinary postage stamp problem, but a quick 10-minute effort on my part failed.

**edit**: (here come the hypotheses!) $a+b+c$ must be odd

If this answer turns out to be negative, or the question turns out to be hard, let's make it easier by increasing the number of coordinates: $(1,\dots,1) \mathbb{N} \setminus \sum \mathbf{v} \mathbb{N}$, where the sum is over all $\mathbf{v}$ with one entry of each of $a$, $b$, $c$, and the rest $0$. Does this modification of the problem start to approach the ordinary coin problem for $a,b,c$?