13 problems

Let $M_1,\ldots,M_n$ be matroids on the same finite ground set $E$, with $\chi(M_i)\leq n$ for every $i=1,\ldots,n$. Suppose $n$ agents have additive valuations $v_1,\ldots,v_n$ of…

Let $M=(E,\mathcal{I})$ be a matroid on the finite set of goods $E$, and let $\chi(M)$ be the least number of independent sets in a partition of $E$. Suppose $\chi(M)\leq n$ and th…

Let $M=(E,\mathcal{I})$ be an independence structure, where $E$ is a finite set and $\mathcal{I}$ is closed under taking subsets. Let $\chi(M)$ be the least number of independent s…

Let $m$ agents and $n$ goods be given, with valuation parameters $v_{\ell i}$ and $V=\sum_{\ell,i}v_{\ell i}$. Let $\epsilon>0$ be arbitrarily small and assume $V\leq\epsilon$. Let…

Equitable distribution conjecture. It should be possible to cut at most $(r-1)m$ cookies and distribute them so that each child has the same amount of each kind of frosting and at…

Let $\mathcal{M}_1$ and $\mathcal{M}_2$ be matroids on the same ground set $E$, and assume that $E$ can be partitioned into $k$ independent sets in each matroid. For a set…

Crew–Narayanan–Spirkl conjecture. There is a partition $[n]=P\cup Q$ into two nonempty sets and a partition $S^1=X\cup X^c$ into two intervals such that

Let an instance consist of $n$ agents and $m$ items. A symEF$(n-1)$ allocation is an allocation satisfying symmetric envy-freeness up to $n-1$ items. SymEF$(n-1)$ existence conject…

Consider instances with a fixed number $n$ of agents and $m$ items, with probability taken over the paper's instance model. A symEF1 allocation is an allocation satisfying symmetri…

Let $n,d$ be positive integers, and let $(\mu^1_1,\dots,\mu^1_n),\dots,(\mu^d_1,\dots,\mu^d_n)$ be $d$ tuples of $n$ absolutely continuous probability measures on $\mathbb{R}^d$. A…

Let $\mathcal{T}$ be a tangle, and let $n$ agents have monotone continuous valuations. The Generalized gap threshold conjecture. A tangle $\mathcal{T}$ guarantees connected envy-fr…

Let $G$ be a biconnected graph. For positive integers $a,b$, say that $G$ is $(a,b)$-linked if, for every pair of disjoint vertex sets $M_1,M_2$ with $|M_1|=a$ and $|M_2|=b$, there…

Universal upper-bound conjecture. For every $n$, every set of $n$ value measures, and every entitlement vector $t$, there exists a $t$-proportional allocation with at most $2n-2$ c…