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Norihide Tokushige

Publications and source records attributed to Norihide Tokushige.

At least 19 recordsLinked to original sources
arXiv2026

A product theorem for $r$-cross intersecting families of subspaces

Let $V$ be an $n$-dimensional vector space over a finite field of order $q$. Let $r\geq 3$, $(r-1)n\geq rk$ and let $\mathcal F_1,\ldots,\mathcal F_r\subset \genfrac{[}{]}{0pt}{}{V}{k}$, where $\genfrac{[}{]}{0pt}{}{V}{k}$ denotes the set of $k$-dimensional subspaces of $V$. Suppose that $F_1\cap\cdots\cap F_r\neq\{0\}$ holds for all $F_i\in\mathcal F_i$, $1\leq i\leq r$. Then we show that $\prod_{i=1}^r|\mathcal F_i|\leq\genfrac{[}{]}{0pt}{}{n-1}{k-1}$, provided $n-k$ is sufficiently large for fixed $q$ and $r$. Moreover, equality holds if and only if there is a common line $L$ such that every family $\mathcal F_i$ consists of all $k$-dimensional subspaces containing the line $L$. One of the main tools of the proof is a junta theorem concerning intersecting linear maps obtained by Ellis, Kindler, and Lifshitz.

math.CO↗
arXiv2026

A semidefinite programming approach to cross $2$-intersecting families

Let $k\geq 2$ and $n\geq 3(k-1)$. Let $\mathcal{F}$ and $\mathcal{G}$ be families of $k$-element subsets of an $n$-element set. Suppose that $|F\cap G|\geq 2$ for all $F\in\mathcal{F}$ and $G\in\mathcal{G}$. We show that $|\mathcal{F}||\mathcal{G}|\leq\binom{n-2}{k-2}^2$, and determine the extremal configurations. This settles the last unsolved case of a recent result by Zhang and Wu (J. Combin. Theory Ser. B, 2025). We also obtain the corresponding result in the product measure setting. Our proof is done by solving semidefinite programming problems.

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arXiv2025

Burning numbers via eigenpolytopes -- Hamming graphs, Johnson graphs, and halved cubes

We give lower and upper bounds on the burning number of Hamming graphs, Johnson graphs, and halved cube graphs. For the lower bounds, we use the fact that $1$-skeletons of the eigenpolytopes of these graphs are isomorphic to the original graphs. Then, we present a dynamic search algorithm performed on the eigenpolytope to find an unburned vertex. This idea was originally used by Alon (Discrete Appl.\ Math.,\ 1992), who determined the burning number of the hypercube graphs.

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arXiv2025

Alon's transmitting problem and multicolor Beck--Spencer Lemma

The Hamming graph $H(n,q)$ is defined on the vertex set $\{1,2,\ldots,q\}^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon (1992) proved that for any sequence $v_1,\ldots,v_b$ of $b=\lceil\frac n2\rceil$ vertices of $H(n,2)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. In this note, we prove that for any $q\geq 3$ and any sequence $v_1,\ldots,v_b$ of $b=\lfloor(1-\frac1q)n\rfloor$ vertices of $H(n,q)$, there is a vertex whose distance from $v_i$ is at least $b-i+1$ for all $1\leq i\leq b$. Alon used a lemma due to Beck and Spencer (1983) which, in turn, was based on the floating variable method introduced by Beck and Fiala (1981) who studied combinatorial discrepancies. For our proof, we extend the Beck--Spencer Lemma by using a multicolor version of the floating variable method due to Doerr and Srivastav (2003).

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arXiv2024

Burning Hamming graphs

The Hamming graph $H(n,q)$ is defined on the vertex set $[q]^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon \cite{Alon} proved that the burning number of $H(n,2)$ is $\lceil\frac n2\rceil+1$. In this note we give a short proof of a fact that the burning number of $H(n,q)$ is $(1-\frac 1q)n+O(\sqrt{n\log n})$ for fixed $q\geq 2$ and $n\to\infty$.

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arXiv2024

Strong stability of 3-wise $t$-intersecting families

Let ${\mathcal G}$ be a family of subsets of an $n$-element set. The family ${\mathcal G}$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in ${\mathcal G}$ is of size at least $t$. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in{\mathcal G}$. For example, if ${\mathcal G}$ consists of all subsets containing a fixed $t$-element set, then it is a $3$-wise $t$-intersecting family with the measure $p^t$. Let $0 0$, and let ${\mathcal G}$ be a $3$-wise $t$-intersecting family. It is known that the measure of ${\mathcal G}$ is at most $p^t$. Suppose, moreover, that ${\mathcal G}$ has the measure at least $(\frac12+δ)p^t$. We show that, by choosing $t$ sufficiently large depending on $δ$, the structure of ${\mathcal G}$ is one of (i) and (ii): (i) every subset in ${\mathcal G}$ contains a fixed $t$-element set, (ii) every subset in ${\mathcal G}$ contains at least $t+2$ elements from a fixed $(t+3)$-element set.

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arXiv2023

The maximum measure of non-trivial 3-wise intersecting families

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called non-trivial $3$-wise intersecting if the intersection of any three subsets in $\mathcal G$ is non-empty, but the intersection of all subsets is empty. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. We determine the maximum measure of non-trivial $3$-wise intersecting families. We also discuss the uniqueness and stability of the corresponding optimal structure. These results are obtained by solving linear programming problems.

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arXiv2023

Non-trivial 3-wise intersecting uniform families

A family of $k$-element subsets of an $n$-element set is called 3-wise intersecting if any three members in the family have non-empty intersection. We determine the maximum size of such families exactly or asymptotically. One of our results shows that for every $ε>0$ there exists $n_0$ such that if $n>n_0$ and $\frac25+ε<\frac kn<\frac 12-ε$ then the maximum size is $4\binom{n-4}{k-3}+\binom{n-4}{k-4}$.

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arXiv2023

The maximum measure of 3-wise t-intersecting families

Let $\mathcal G$ be a family of subsets of an $n$-element set. The family $\mathcal G$ is called $3$-wise $t$-intersecting if the intersection of any three subsets in $\mathcal G$ is of size at least $t$. For a real number $p\in(0,1)$ we define the measure of the family by the sum of $p^{|G|}(1-p)^{n-|G|}$ over all $G\in\mathcal G$. We prove that if $t\geq 15$ and $p\leq 2/(\sqrt{4t+9}-1)$ then $p^t$ is the maximum measure of $3$-wise $t$-intersecting families, and the bound for $p$ is sharp. We also present the corresponding stability result for shifted families.

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arXiv2022

Counting cliques in a random graph

We show that the expected number of cliques in the Erdős-Rényi random graph $G(n,p)$ is $n^{\frac1{-2\log p}(\log n-2\log\log n+O(1))}$.

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arXiv2021

Application of hypergraph Hoffman's bound to intersecting families

Using the Filmus-Golubev-Lifshitz method to bound the independence number of a hypergraph, we solve some problems concerning multiply intersecting families with biased measure. Among other results we obtain a stability result of a measure version of the Erdos-Ko-Rado theorem for multiply intersecting families.

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arXiv2019

Avoiding a star of three-term arthmetic progressions

We provide an upper bound of the size of a subset A of F_p^n that does not admit a k-star of 3-APs (three-term arithmetic progressions). Namely, the subset A is assumed to contain no configuration of k 3-APs, sharing the middle term, such that all 2k+1 terms are distinct. In the proof, we adapt a new method in the recent work of Sauermann.

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arXiv2019

Avoiding a shape, and the slice rank method for a system of equations

Fix a vector space over a finite field and a system of linear equations. We provide estimates, in terms of the dimension of the vector space, of the maximum of the sizes of subsets of the space that do not admit solutions of the system consisting of more than one point. That from above is derived by slice rank method of Tao; to obtain one from below, we define the notion of 'dominant reductions' of the system. Furthermore, by adapting a recent argument of Sauermann, we make an estimation of the maximum of the sizes of subsets that are 'W shape'-free, that means, there exist no five distinct points forming two overlapping parallelograms.

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arXiv2019

AK-type stability theorems on cross t-intersecting families

Two families, ${\mathcal A}$ and ${\mathcal B}$, of subsets of $[n]$ are cross $t$-intersecting if for every $A \in {\mathcal A}$ and $B \in {\mathcal B}$, $A$ and $B$ intersect in at least $t$ elements. For a real number $p$ and a family ${\mathcal A}$ the product measure $μ_p ({\mathcal A})$ is defined as the sum of $p^{|A|}(1-p)^{n-|A|}$ over all $A\in{\mathcal A}$. For every non-negative integer $r$, and for large enough $t$, we determine, for any $p$ satisfying $\frac r{t+2r-1}\leq p\leq\frac{r+1}{t+2r+1}$, the maximum possible value of $μ_p ({\mathcal A})μ_p ({\mathcal B})$ for cross $t$-intersecting families ${\mathcal A}$ and ${\mathcal B}$. In this paper we prove a stronger stability result which yields the above result.

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arXiv2016

A semidefinite programming approach to a cross-intersection problem with measures

We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ratio bound on the independence number of a graph. As an application, we solve a problem on the maximum measures of cross-intersecting families of subsets with two different product measures, which is a generalized measure version of the Erdős-Ko-Rado theorem for cross-intersecting families with different uniformities.

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arXiv2016

Multiply union families in $\mathbb{N}^n$

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) $n=r+1$.

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