The spectrum of P_(k,10-k)^(5)-designs

NEW

Problem: Reconstructed statement: in the finite $5$-uniform hypergraph-design sense, $P^{(5)}(k,10-k)$ is the two-edge $5$-uniform hypergraph $\{A,B\}$ with $|A|=|B|=5$, $|A\cap B|=k$, hence $|A\cup B|=10-k$. A $P^{(5)}(k,10-k)$-design of order $v$ is a partition of all $5$-subsets of a $v$-set into such pairs. The conjecture claims that for $k=1,2$ this exists iff $v\ge 10-k$ and $v\equiv1,3\pmod8$.

Result: The conjecture is false. The correct ordinary spectrum for $k=1,2$ is

equivalently

Necessity: each block contains two $5$-edges, so $\binom v5$ must be even; also a copy uses $10-k$ vertices.

Sufficiency: let $\Gamma_{v,k}$ have as vertices the $5$-subsets of a $v$-set $X$, with $A\sim B$ iff $|A\cap B|=k$. A perfect matching in $\Gamma_{v,k}$ is exactly a $P^{(5)}(k,10-k)$-design.

The graph is vertex-transitive under $\operatorname{Sym}(X)$. It is connected: every Johnson-adjacent pair $A=S\cup\{a\}$, $B=S\cup\{b\}$, $|S|=4$, is joined by a length-two path. For $k=1$, since $v\ge9$, choose three points outside $A\cup B$ and take $C=\{a,b,x,y,z\}$. For $k=2$, since $v\ge8$, choose $s\in S$ and two points outside $A\cup B$, and take $C=\{s,a,b,x,y\}$. Thus $\Gamma_{v,k}$ is connected. Since $|V(\Gamma_{v,k})|=\binom v5$ is even, the theorem that every connected vertex-transitive graph of even order has a perfect matching applies.

Finally, by Lucas’ theorem mod $2$, $\binom v5$ is odd exactly when $v\equiv5,7\pmod8$.

Thus, for example, $k=2,\ v=8$ gives a $P^{(5)}(2,8)$-design, because $\binom85=56$ is even, but $8\not\equiv1,3\pmod8$. This disproves the stated “only if” direction.

Citation: Uses the standard theorem that every connected vertex-transitive graph of even order has a perfect matching; see Lovász and Plummer, Matching Theory, North-Holland, 1986. No cited source is known here to already state this exact corrected spectrum.