An \(\mathrm{STS}(v)\) has \(r=6\) and \(b=26\). Is \(v=13\) the only possible value, and what about \(v=12\)? Justify using the bounds on \(r\) and \(b\).
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B. Yes, \(v=13\) is unique and \(v=12\) is invalid because \(v\) must be odd
From \(r=\frac{v-1}{2}=6\) we get \(v=13\), and \(v\) must be odd. That \(\frac{12\cdot 11}{6}=22\) is an integer is not enough: \(v=12\) fails parity, which rules it out.
A Steiner triple system \(\mathrm{STS}(v)\) is a block design in which \(v\) points are grouped into triples so that every pair of distinct points lies in exactly one triple.