Odd-order additive gap

Determine the value of \(\rho_{q+1,q-h+3}\) for every odd prime power \(q\) and integer \(2\le h<q\), and establish whether a non-projective finite linear intersecting \((q+1)\)-uniform hypergraph can attain the upper bound \((q+1)/h\) despite the nonexistence of nontrivial maximal \(h\)-arcs in \(PG(2,q)\), or whether the truncated-plane value \(q/h\) is optimal in some odd-order cases.

Background

For $r=q+1$, the paper defines $h=q-k+3$ and studies the efficiency parameter $\rho_{q+1,k}$, the supremum of the ratio between the number of edges and the minimum size of an $h$-fold transversal over finite linear intersecting $(q+1)$-uniform hypergraphs. The universal upper bound is $(q+1)/h$.