27 problems

Kalai–Sarkaria conjecture. If $K$ admits an embedding into $S^{r}$, then

Let $t\leq r$ be two natural numbers. A family of faces is $t$-intersecting if every pair of faces in it has intersection of cardinality at least $t$. Let $\Delta$ be a simplicial…

Let $L\subseteq K$ be simplicial complexes, let $\Delta(K)$ denote the algebraically shifted complex of $K$, and let … be the dimension of the $j$th reduced relative homology group…

Kalai's conjecture. For every such shifted order ideal,

Let $\Gamma$ be a simplicial complex, and let $b^{\operatorname{\Delta}^s}_{i,r}(\Gamma)$ and $b^{\operatorname{\Delta}^e}_{i,r}(\Gamma)$ denote the corresponding invariants for sy…

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring with each $\deg x_i=1$, and let $\Delta$ be a simplicial complex on $[n]=\{1,\ldots,n\}$. Write $I_\Delta\subset S$ for its Stanley…

Let $K$ be a simplicial complex. Write $b\Delta(K)d$ for its algebraic shifting and let $\Sigma(K)$ denote its suspension, the join of $K$ with the shifted simplicial complex consi…

Initial lex-segment characterization. The hypergraph $H$ is matroidal if and only if its hyperedges form an initial segment with respect to the lexicographic order.

Characteristic-zero exterior shifting conjecture.

Barycentric subdivision conjecture. If $K$ is a surface triangulation, then $\Delta^{ex}(\operatorname{sd}(K))$ is a homology lex-segment.

Uniform surface conjecture. For every fixed genus $g$, a.a.s.,

Lex-segment prevalence conjecture. As $n$ tends to infinity,

Lexicographic edge conjecture. If $\sigma<_{\mathrm{lex}}\sigma'$, then

Bulavka–Nevo–Peled conjecture. The triangle $\{1,3,n\}$ belongs to $\Delta(K)$ for every triangulation $K$ of a compact surface without boundary.

A triangulation of a compact connected surface without boundary is a simplicial complex representing the surface; removing a single triangle means deleting that triangular face and…

Let $H(k,r,n):=\{F\in\binom{[n]}{k}:F\cap [r]\neq\varnothing\}$ and abbreviate $H(r,n):=H(3,r,n)$. For a $k$-uniform hypergraph $H$ on $[n]$, let $\operatorname{rank}_r(H)$ be the…

Let $H$ be a $3$-uniform hypergraph on $[n]$ such that every four vertices span an edge, and let $H^{<_{L}}$ denote its exterior algebraic shift with respect to the lexicographic o…

Let $H$ be a $3$-uniform hypergraph on $[n]$ such that every four vertices span an edge. Let $C(n)$ be … and let $<_{c}$ be a term order on $\binom{[n]}{3}$ in which $C(n)$ is an i…

Let $[n]=\{1,\ldots,n\}$, and let $C(n)$ be the family … For a $3$-uniform hypergraph $H$, say that $H$ dominates $C(n)$ when the corresponding rows and columns of the third compou…

Algebraic and symmetric shifting conjecture. For all $i,j$,

Let $K$ be a $d$-dimensional balanced simplicial complex, and let $<$ be a $(3,\ldots,3)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced linkless…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,\ldots,2)$-admissible order. Let $K^{b,<}$ denote the corresponding balanced shift. Balanced PL-embeddability co…

Let $K$ be a $d$-dimensional balanced complex, and let $<$ be a $(2,2,\ldots,2)$-admissible order, meaning that the least two vertices of each color class form an initial segment o…

Let $K$ be a $d$-dimensional balanced simplicial complex, meaning that its vertices are colored with $d+1$ colors so that every edge has vertices of different colors. Let $K^b$ den…

Let $K$ be a $2$-dimensional simplicial complex, and let $K^s$ denote its symmetric algebraic shifting. Symmetric-shifting conjecture. If $K$ is embeddable in $\mathbb{R}^4$, then…