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

Existence of a block design

Let $\ell$ be an integer parameter. I want to ask the existence of the following design: There is a universal constant $\beta < 1$ such that for all sufficiently large $\ell$, the following holds: The universe size $d = O(\ell)$. There are $\ell^2$ sets in the design, $S_1, \dots, S_{\ell^2} \subseteq [d]$. $(\forall i \in … Read more

Integer solution

For every prime p, does there exists integers x1, x2 and x3 (0≤x1,x2,x3≤⌊cp1/3⌋ and c is some large constant) such that p−12−⌊2cp1/3⌋≤f(x1,x2,x3)≤p−12, where, f(x1,x2,x3)=x1+x2+x3+2(x1x2+x2x3+x3x1)+4x1x2x3. Answer I doubt that the lower bound p−12−2cp1/3 holds for all p. Here is a proof for the weaker bound p−12−cp1/2. First of all, the inequality p−12−cp1/2≤f(x1,x2,x3)≤p−12 is essentially equivalent to … Read more

The degree/diameter problem for even girth graphs starting with upper bound

I posted this on stackexchange but due to a lack of response there I am posting here. Let G be a graph with girth g, minimal degree δ, maximal degree Δ, and diameter D. Define n0(g,δ):={1+δ+δ(δ−1)+⋯+δ(δ−1)g−32, if g is odd2+2(δ−1)+⋯+2(δ−1)g2−1, if g is even.. Suppose the girth g is odd. It can be shown that the number of vertices of G … Read more

Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non overlapping hyperspheres whose centers are the closest to $p$ under the Euclidean metric (although I’m eventually interested in exploring other metrics as … Read more

(Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know: Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$. There are crystal operators, $e_i$, $f_i$ that preserves the shape, but converts one box of content $i$ to a box with content $i+1$ … 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