We recall the concept of the dimension of a finite poset $P$, and the longstanding conjecture that for all finite nonempty posets $P$ and $Q$, $\dim(P\times Q)\geq\dim(P)+\dim(Q)-2.$ We then note two other plausible inequalities, either of which would imply that one. In the final section, writing $P\preccurlyeq P'$ if, for all $Q,$ $\dim(P\times Q)\leq\dim(P'\times Q),$ and writing $P\approx P'$ if $P\preccurlyeq P'$ and $P'\preccurlyeq P,$ we note some results and questions concerning these relations.
Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Mart\'{i}nez-Celis, Morawski and \'Swierczy\'nska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..
Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $\chi \in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<\chi(1)_p\le m(P)$, giving one inequality of the Eaton-Moret\'o conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moret\'o conjecture for principal blocks.
Asier Arranz, J. Gómez-Serrano, Gabriel Navarro et al.· 1 citation
Let $X$ be a smooth, projective, geometrically connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every action of a finite $p$-group on $X$ factors through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer properties of representations of finite $p$-groups to their actions on $X$. We deduce a fixed-point theorem which, unlike previously known results of this kind, is sensitive to the arithmetic of the base field. We also obtain a bound on the orders of cyclic $p$-subgroups of the Cremona groups: for instance $\operatorname{Cr}_n(\mathbb{Q})$ contains no element of order $p^2$ when $p \ge n+2$. The method is based on the following observation, of independent interest. For an affine algebraic group $G$ over a field of characteristic zero, $\operatorname{ed}_p(G) + \dim G$ is the least dimension of a smooth projective variety $Y$ with a generically free $G$-action such that the degree map $\operatorname{CH}_G(Y) \to \mathbb{F}_p$ is nonzero. A key input for our result is Karpenko and Merkurjev's computation of the essential $p$-dimension of $p$-groups.
Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$-module. Assume $M$ has rank $r>0$. We show that if $M$ is NOT Cohen-Macaulay then $\mu_d(\mathfrak{m}, M)>r$. If further $A$ is unmixed and $\mu_n(\mathfrak{m}, M) \leq 1$ for some $n \geq d$ then we prove $\text{injdim} \ M<\infty$ and $A$ is Cohen-Macaulay.
In this note, we prove that an arbitrary product of $W$-spaces is a $\kappa$-Fr\'{e}chet-Urysohn space. We also show that the boundary tightness of a product $X \times Y$ is at most $\kappa$ provided that $X$ is compact with tightness $t(X) \leq \kappa$ and $Y$ has boundary tightness $tb(Y) \leq \kappa$. The first result fully resolves, and the second partially addresses, two questions recently raised by Tkachuk.
Mikołaj Krupski, Kacper Kucharski· 0 citations
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