Related papers: The four-way intersection problem for latin square…

We consider the complexity of the recognition problem for two families of combinatorial structures. A graph $G=(V,E)$ is said to be an intersection graph of lines in space if every $v\in V$ can be mapped to a straight line $\ell (v)$ in…

Can we color the $n^3$ cells of an $n\times n\times n$ cube $L$ with $n^2$ colors in such a way that each layer parallel to each face contains each color exactly once and that the coloring is symmetric so that $L_{ij\ell}=L_{j\ell…

We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families $\mathcal{F}$ of $k$-element subsets of $[n]:=\{1,\ldots, n\},$ such that every element of $[n]$ lies in the…

A classical question in combinatorics is the following:\ given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of \textbf{$\epsilon$-dense partial Latin squares}:\ partial…

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each…

The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex…

Latin squares and hypercubes are combinatorial designs with several applications in statistics, cryptography and coding theory. In this paper, we generalize a construction of Latin squares based on bipermutive cellular automata (CA) to the…

We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced…

We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no…

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu'…

We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of…

We determine necessary conditions for ample divisors in arbitrary genus as well as for very ample divisors in genus 2 and 3. We also compute the intersection numbers $\lambda^9$ and $\lambda_{g-1}^3$ in genus 4. The latter number is…

A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The…

It was shown by van Rees \cite{vR} that a latin square of order $n$ has at most $n^2(n-1)/18$ latin subsquares of order $3$. He conjectured that this bound is only achieved if $n$ is a power of $3$. We show that it can only be achieved if…

In 1990, Kolesova, Lam and Thiel determined the 283,657 main classes of Latin squares of order 8. Using techniques to determine relevant Latin trades and integer programming, we examine representatives of each of these main classes and…

We calculate intersection forms of all 4-dimensional almost-flat manifolds

We show that every pair of longest paths in a $k$-connected graph on $n$ vertices intersect each other in at least $(8k-n+2)/5$ vertices. We also show that, in a 4-connected graph, every pair of longest paths intersect each other in at…

We present the number of totally symmetric quasigroups (equivalently, totally symmetric Latin squares) of order 16, as well as the number of isomorphism classes, and extend previously published results to include information on the number…

Consider the Birkhoff polytope of n by n doubly-stochastic matrices. As the Birkhoff-von Neumann theorem famously states, its vertex set coincides with the set of all n by n permutation matrices. Here we seek a higher-dimensional analog of…

We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $\Theta(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list…