## How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more

## A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I have below. Let G=(V,E) be a graph (or H=(V,E) a hypergraph). Let E=A1⊎A2⊎⋯⊎Al be a partition. I am looking at coloring of … Read more

## Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety V with a symplectic structure and a torus action, there is a moment map μ:V→Lie(T)∗. Note that the dimension of T could be much smaller than the dimension of V. How much can we say about the fibers of this moment map μ? Any references? I am most … Read more

## Weighted maximal number of disjoint singly-generated ideals in the divisibility poset for {1,2,…,n}\{1,2,\ldots,n\}

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset Pn of the set of natural numbers {1,…,n} is considered. I have a somewhat dual question. Consider a “packing” of disjoint singly generated ideals (upsets) Cx where Cx={x,y1x,y1y2x,…}∈{1,2,…,n} with yi natural numbers greater than 1. To be precise, an ideal C … Read more

## Kruskal-Katona for homocyclic groups?

I need a version of the Kruskal-Katona theorem (better still, of the Lovasz “approximate” version thereof) for the elementary abelian / homocyclic groups, in the following spirit: What is the smallest possible size of the shadow of a subset A⊂(Z/mZ)n given the size N:=|A| and the common weight k of all the elements of A? … Read more