arXiv subjects

Ohad Sheinfeld

Publications and source records attributed to Ohad Sheinfeld.

5 recordsLinked to original sources

arXiv2026

A Complete Intersection Theorem for Large Permutation Groups

A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size of a $t$-intersecting family in $S_n$ is obtained by one of the families $\mathcal{F}_{n,t,r}=\{\sigma \in S_n: |\mathrm{Fixed}(\sigma) \cap \{1,2,\ldots,t+2r\}|\geq t+r\}$, where $\mathrm{Fixed}(\sigma)$ is the set of fixed points of $\sigma$. This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977).

math.CO↗

arXiv2026

Forbidden Intersection Theorems for Matrix Spaces

A family of $m \times n$ matrices $\mathcal{F} \subseteq \mathbb{F}_q^{m \times n}$ is {$(t-1)$-intersection-free} if $\dim \ker(A-B) \neq t-1$ for all $A,B \in \mathcal{F}$. A \emph{forbidden $(t-1)$-intersection problem} for a collection of matrices asks for the size and structure of extremal $(t-1)$-intersection-free families within that collection. We solve this problem in $\mathrm{GL}(n,q)$ for all pairs $(n,t)$ such that $t 0$. We also give Frankl--R\"odl-type constructions showing that this range of $t$ is almost the best possible: we show that for values of $t>n/2$ the extremal behavior changes and no clean analogue is expected. Our proof builds upon recent global hypercontractivity results for matrix spaces due to Evra, Kindler, and Lifshitz, and broadly applies to any sufficiently dense class of matrices.

math.CO↗

arXiv2025

The Forbidden Cross Intersection Problem for Permutations

We prove the following, for a universal constant $c>0$. Let $n \in \mathbb{N}$ and $1 \leq t 0$, the statement fails for $t=(1+\epsilon)\frac{n}{\log_2 n}$ and all $n>n_0(\epsilon)$. This solves the cross-intersection variant of the Erd\H{o}s-S\'{o}s forbidden intersection problem for permutations. The best previously known result, by Kupavskii and Zakharov (Adv.~Math., 2024), obtained the same assertion for $t \leq \tilde{O}(n^{1/3})$. We obtain our result by combining two recently introduced techniques: hypercontractivity of global functions and spreadness.

math.CO↗

arXiv2023

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as \emph{normal} sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and $A$ is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon}}$, then $A^2 \supseteq A_n$. This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$. Our proof strategy combines two types of techniques. The first is `traditional' techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck--Shalev, Larsen--Shalev, and more recently, Larsen--Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

math.GR↗

arXiv2023

On $t$-Intersecting Families of Permutations

We prove that there exists a constant $c_0$ such that for any $t \in \mathbb{N}$ and any $n\geq c_0 t$, if $A \subset S_n$ is a $t$-intersecting family of permutations then$|A|\leq (n-t)!$. Furthermore, if $|A|\ge 0.75(n-t)!$ then there exist $i_1,\ldots,i_t$ and $j_1,\ldots,j_t$ such that $\sigma(i_1)=j_1,\ldots,\sigma(i_t)=j_t$ holds for any $\sigma \in A$. This shows that the conjectures of Deza and Frankl (1977) and of Cameron (1988) on $t$-intersecting families of permutations hold for all $t \leq c_0 n$. Our proof method, based on hypercontractivity for global functions, does not use the specific structure of permutations, and applies in general to $t$-intersecting sub-families of `pseudorandom' families in $\{1,2,\ldots,n\}^n$, like $S_n$.

math.CO↗