5 problems

Dawar's conjecture. For every finitely axiomatizable class $K$ of rigid structures, some formula in the fixed-point extension of first-order logic defines a linear order in $K$.

McColm's conjecture. If every $(\operatorname{FO}+\operatorname{LFP})$ formula is equivalent to a first-order formula in $K$, then every positive elementary induction is bounded in…

Merge-elimination conjecture. Every instance of $\textsc{Merge}$ that could be needed to construct $\mathit{RK}_\mu^\mathbb{F}$-proofs corresponding to $\mathit{K}_\mu$-proofs is s…

Let $\mathscr C$ be an infinite class of finite ordered structures. Kolaitis's ordered conjecture. There is a sentence $\varphi\bin \text{\sf{LFP}}$ that is not equivalent on…

Let $\mathcal{C}$ be a family of finite structures. The family $\mathcal{C}$ is non-proficient if it does not have the $LFP$-strict order property, equivalently, if it is not profi…