Bounds for column-design covering width

Bound the quantity ω_c(r, c, v), defined as the least width of the interval of pair-covering numbers in a block design with v blocks and c points, point replication number r, and block sizes floor(rc/v) or ceil(rc/v), in terms of r, c, and v.

Background

The column design of a near triple array is a block design whose point occurrences are fixed, whose block sizes are one or two consecutive values, and whose pair-covering numbers are one or two consecutive values. The authors formulate a relaxed version for generalized triple arrays by allowing a wider interval of pair-covering numbers.

The function ω_c(r,c,v) isolates the column-design component of the generalized triple-array problem and asks for parameter-dependent bounds on the minimum possible covering-number width.

References

As an intermediate goal, we pose a related problem for the column design of a generalized triple array. Let $\omega_c(r, c, v)$ be the smallest integer $\omega_c$ such that there exists an integer $x$ and a block design with $v$ blocks and $c$ points, with each point occurring in $r$ blocks, each block containing $\lfloor\frac{rc}{v}\rfloor$ or $\lceil\frac{rc}{v}\rceil$ points, and each pair of points covered by at least $x$ and at most $x + \omega_c - 1$ blocks. How can $\omega_c(r, c, v)$ be bounded in terms of $r, c, v$?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section 7, Concluding remarks, intermediate problem for column designs

How can $\omega_c(r, c, v)$ be bounded in terms of $r, c, v$?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section “Concluding remarks,” Question environment beginning “Let ω_c(r, c, v)”