The site writes "each pair of $y_i,y_j$ is adjacent to a unique $z_i$", an index that does not say on its face whether different pairs may share a $z$. The sources fix one vertex for each pair. Erdős's source [Er71] (item 15, pp. 103--104) defines the graph, there called $G_k$, as a vertex $x$ joined to $y_1,\ldots,y_k$ with each pair of the $y$'s joined to its own vertex $z$ (see the source card). Füredi's paper on this question [Fu91] (printed p. 76) defines the same graph $L^k$, the lowest three levels of the Boolean lattice, with one vertex $x_{ij}$ for each pair $1\leq i<j\leq k$ joined to exactly $y_i$ and $y_j$, and the site's commentary credits that theorem with the answer. The site's own vertex list, with $\binom{k}{2}$ vertices $z$, has one for each pair. The precise Statement takes this reading: the vertex $z_{\{i,j\}}$ of the pair $\{i,j\}$, with edges $xy_i$, $y_iz_{\{i,j\}}$ and $y_jz_{\{i,j\}}$ and no others. The formal-conjectures statement (see Formalization) uses the same graph.