We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the maximum order of a minimally transitive permutation group of degree $n$ is $2^{\Theta(n)}$. For completeness, we also present Pyber's previously unpublished proof of his upper bound. We further deduce that the numbers of labelled vertex-transitive graphs and digraphs of order $n$ are both $2^{\Theta(n\log(n))}$, and discuss the implications of our results for approaches to the McKay--Praeger conjecture.
A family of permutations in $S_n$ is called even-intersecting if every two distinct members agree in an even number of positions. Let $M(n)$ denote the maximum size of such a family. For even $n$, we prove that $$n!!\leq M(n)\leq e^{\frac{n}{2}+o(n)}n!!,$$ improving the bound obtained from a theorem of Cameron, Deza an...
Anirban Banerjee, Abisek Dewan, R. Mishra· 0 citations
Let $f(n)$ denote the number of isomorphism classes of groups of order $n$. We prove, uniformly over all primes $p$, that \[ \log_p f(p^m)=\frac{2}{27}m^3-\frac49m^2+O(m), \] with an absolute implied constant. This improves the classical upper bounds of Sims (\emph{Proc. Lond. Math. Soc.}, 1965) and Newman--Seeley and...
We prove the following two upper bounds for ordered Ramsey numbers: (1) Every ordered forest $F$ on $n$ vertices satisfies $R_{<}(F,F)=O(n^{1+\lceil\log\chi_{<}(F)\rceil})$. This in particular answers a question of Geneson, Holmes, Liu, Neidinger, Pehova and Wass. (2) There is a function $f$ such that, for every fixed...
We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let $\omega(G)$ denote the largest size of a pairwise noncommuting subset of a group $G$, let $a(G)$ be the least size of an abelian cover, and define $h(n)=\sup\{a...
The rows and columns of the character table of the symmetric group $S_n$ are both naturally indexed by partitions of $n$. Let $D(n)$ denote the number of conjugacy classes of $S_n$ whose column contains no zero entry. The identity column is always zero-free, so $D(n)\geq 1$. It is known that $D(n)\ll n^2$. We prove tha...
Colin Defant, S. Hariharan, Kenny Lau et al.· 0 citations
We show that there exists a constant $\epsilon>0$ such that if $C$ is a conjugacy class in $A_n$ of size at least $|A_n|^{1-\epsilon}$, then $A_n \setminus \{1\} \subseteq C^2$. This proves the robust version of Thompson's conjecture for the alternating group, conjectured by Shalev (Annals of Math., 2009). Our proof co...
Nathan Keller, Noam Lifshitz, Avichai Marmor et al.· 1 citation
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