Is there a non-degenerate 3-design where the number of blocks equals the number of points? Non-degenerate in this context means that a point is incident with at least 2 and at most #blocks-2 blocks.
1 Answer
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No. If $b=v$ and $t>2$ the block design equations give that either $k=v$ (in which case there may only be one block and the design is trivial) or $k=v-1$ so that the design is also trivial (see Lemma 2.6 in the paper of Hughes cited below). This is why only quasi-symmetric $t$-designs are discussed in texts when $t>2$.
Hughes, D. R. On $t$-designs and groups. Amer. J. Math. 87 (1965), 761-778.
