6 problems

Let $L$ be a Latin square of order $n\geq 3$, and let $S$ be a subset of its entries with $|S|\leq \left\lfloor n^2/4\right\rfloor$. For a Latin trade $T$ in $L$, let $R(T)$,…

Pair-covering conjecture. Every pair of entries in $B_p$ is contained in a Latin trade of size $d_p$.

Strong defining-set conjecture. Removing any entry from $B_p$ results in a minimum $(d_p-1)$-strong defining set.

Prime-order Latin trade conjecture. The smallest Latin trade size in $L$ is no larger than the smallest Latin trade size in $B_p$.

Logarithmic distance conjecture. For each Latin square $L$ of order $n$,

Let $(W,B)$ be a spherical latin bitrade, and let $\mathcal{A}_W^*$ and $\mathcal{A}_B^*$ denote the canonical abelian groups into which the white and black trades embed, respectiv…