arXiv subjects
Publications and source records attributed to Norihide Tokushige.
Let $A\subset \mathbb{N}^{n}$ be an $r$-wise $s$-union family, that is, a family of sequences with $n$ components of non-negative integers such that for any $r$ sequences in $A$ the total sum of the maximum of each component in those sequences is at most $s$. We determine the maximum size of $A$ and its unique extremal configuration provided (i) $n$ is sufficiently large for fixed $r$ and $s$, or (ii) $n=r+1$.
Ahlswede and Khachatrian's diametric theorem is a weighted version of their complete intersection theorem, itself an extension of the $t$-intersecting Erdős-Ko-Rado theorem. Their intersection theorem says that the maximum size of a family of subsets of $[n] = \{1, \dots, n\}$, every pair of which intersects in at least $t$ elements, is the size of certain trivially intersecting families proposed by Frankl. We address a cross intersecting version of their diametric theorem. Two families $\mathcal{A}$ and $\mathcal{B}$ of subsets of $[n]$ are {\em cross $t$-intersecting} if for every $A \in \mathcal{A}$ and $B \in \mathcal{B}$, $A$ and $B$ intersect in at least $t$ elements. The $p$-weight of a $k$ element subset $A$ of $[n]$ is $p^{k}(1-p)^{n-k}$, and the weight of a family $\mathcal{A}$ is the sum of the weights of its sets. The weight of a pair of families is the product of the weights of the families. The maximum $p$-weight of a $t$-intersecting family depends on the value of $p$. Ahlswede and Khachatrian showed that for $p$ in the range $[\frac{r}{t + 2r - 1}, \frac{r+1}{t + 2r + 1}]$, the maximum $p$-weight of a $t$-intersecting family is that of the family $\mathcal{F}^t_r$ consisting of all subsets of $[n]$ containing at least $t+r$ elements of the set $[t+2r]$. In a previous paper we showed a cross $t$-intersecting version of this for large $t$ in the case that $r = 0$. In this paper, we do the same in the case that $r = 1$. We show that for $p$ in the range $[\frac{1}{t + 1}, \frac{2}{t + 3}]$ the maximum $p$-weight of a cross $t$-intersecting pair of families, for $t \geq 200$, is achieved when both families are $\mathcal{F}^t_1$. Further, we show that except at the endpoints of this range, this is, up to isomorphism, the only pair of $t$-intersecting families achieving this weight.
Two families $\mathcal{A}$ and $\mathcal{B}$, of $k$-subsets of an $n$-set, are {\em cross $t$-intersecting} if for every choice of subsets $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $|A \cap B| \geq t$. We address the following conjectured cross $t$-intersecting version of the Erd\H os--Ko--Rado Theorem: For all $n \geq (t+1)(k-t+1)$ the maximum value of $|\mathcal{A}||\mathcal{B}|$ for two cross $t$-intersecting families $\mathcal{A}, \mathcal{B} \subset\binom{[n]}{k}$ is $\binom{n-t}{k-t}^2$. We verify this for all $t \geq 14$ except finitely many $n$ and $k$ for each fixed $t$. Further, we prove uniqueness and stability results in these cases, showing, for instance, that the families reaching this bound are unique up to isomorphism. We also consider a {\em $p$-weight} version of the problem, which comes from the product measure on the power set of an $n$-set.
Arrangement theory plays an essential role in the study of the unfolding model used in many fields. This paper describes how arrangement theory can be usefully employed in solving the problems of counting (i) the number of admissible rankings in an unfolding model and (ii) the number of ranking patterns generated by unfolding models. The paper is mostly expository but also contains some new results such as simple upper and lower bounds for the number of ranking patterns in the unidimensional case.