10 problems

For $d\geq1$, let $C_n$ be the collection of odd, transitive-symmetric functions $f:\{-1,1\}^n\to\{-1,1\}$. Let $\widehat f(S)$ denote Fourier coefficients, and let…

Balanced-subset family conjecture. For each $k$ and all sufficiently large $n$, there is a family $\mathcal{F}$ of $n$-subsets of $[2n]$ such that:

Let $P$ be the set of players and let $\mathcal{K}$ be a coalition structure, with core stability meaning that no subset $S \subseteq P$ can deviate to form a new coalition and mak…

Lower-bound optimality conjecture. If $\alpha \geq \frac{2}{k+1}$, then an $\alpha$-undominated set of size $k$ always exists.

Non-normality conjecture. Asymptotically almost surely, MUCDs are not normal.

Disconnectedness conjecture. Asymptotically almost surely, MUCDs are not connected.

Irreducibility conjecture. MUCDs are asymptotically almost surely irreducible.

ADO optimality conjecture. For any fixed set of an odd number of distinct abilities for a jury, the ordering that produces the highest reliability is the ADO.

The conjectured bounds. The number $\mathcal{N}(m)$ satisfies

Let a voter distribution have a minority voter population, let $\mathsf{ClusP}$ denote its clustering measure, and let $\mathbb{E}(\mathsf{Rep}_\Delta)$ denote the expected represe…