arXiv subjects

Abisek Dewan

Publications and source records attributed to Abisek Dewan.

3 recordsLinked to original sources

arXiv2026

Even-Intersecting Families of Permutations

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 and Frankl (1987) by an exponential factor. This problem may be viewed as a permutation analogue of the classical Eventown problem for set systems. For odd $n$, we give a construction yielding $M(n)\geq n^2/4$. We further extend this construction to obtain $$ M(n)\geq\bigl(n-\sqrt{n-1}\bigr)^2 ,$$ whenever $n=(q+1)^2+1$ and $q$ is an odd prime power. The latter bound asymptotically matches the upper bound $M(n)\leq(n-1)^2+1$ obtained by Cameron, Deza and Frankl.

math.CO↗

arXiv2026

On Ramsey goodness of $K_{2,n}$ versus cycles

A graph $G$ is called $H$-good if $R(G,H)=(|G|-1)(χ(H)-1)+σ(H)$, where $σ(H)$ denotes the size of the smallest color class in a $χ(H)$-coloring of $H$. In Ramsey theory, it is an interesting problem to study whether a graph $G$ is $H$-good or not. In this article, we study the Ramsey goodness of the pair $(K_{2,n},C_m)$, which naturally lies between the classical star-cycle and book-cycle problems. We prove that \begin{equation*} R(K_{2,n},C_{\{m,m+1\}})=m+1. \end{equation*} for all $m\ge 2n+1$, and consequently establish that \begin{equation*} R(K_{2,n},C_{m})=m+1. \end{equation*} for all $m\ge 3n+4$. This proves that $C_m$ is $K_{2,n}$-good in this range and improves a particular case of a result on the Ramsey goodness by Pokrovskiy and Sudakov. Further, we provide a construction of a graph that disproves the $C_{m}$-goodness of $K_{2,n}$ for all even $m$ satisfying $n\geq m+2$.

math.CO↗

arXiv2026

On Ramsey number of $K_{2,n}$ versus even cycles

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest integer $N$ such that every graph $Γ$ on $N$ vertices contains $G$ or its complement $\overlineΓ$ contains $H$ as a subgraph. In graph Ramsey theory, the star-cycle Ramsey number is well-studied throughout the years. Whereas the Ramsey number of $K_{2,n}$ versus cycle is challenging to determine due to increased structural complexity. In this article, we have obtained an exact value of the Ramsey number $R(K_{2,n}, C_{m})$ for even $m\in [n, 2n-4008]$ and $n\geq 4516$. In particular, we show that $$R(K_{1,n}, C_{m})= R(K_{2,n}, C_{m})$$ for all even $m\in [n, 2n-4008]$ and $n\geq 4516$. This leads to an interesting question: For fixed $t$, does there exist $n_0(t)\in \mathbb{N}$ such that $R(K_{1,n}, C_m)=R(K_{t,n}, C_m)$ for all $n \geq n_0(t)$ and for a given range of even $m$?

math.CO↗