Some results on $\lambda$-design conjecture

Yadav, Ajeet Kumar

doi:10.22108/toc.2024.139121.2102

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	Some results on $\lambda$-design conjecture
Transactions on Combinatorics
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 23 مرداد 1403 اصل مقاله (481.97 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2024.139121.2102
نویسنده
Ajeet Kumar Yadav^*
Department of Mathematics, St. Gonsalo Garcia College, University of Mumbai, India
چکیده
Let $v$ and $\lambda$ be integers with $0<\lambda<v$. A $\lambda$-design $D$ is a pair $(X, \mathcal{A})$, where $X$ is a finite set with $v$ elements called points and $\mathcal{A}$ is a family of subsets of $X$ called blocks, with $\|\mathcal{A}\|=\|X\|$ such that (1) for all $B_i, B_j\in \mathcal{A},$ $i\neq j,$ $\|B_i\cap B_j\|=\lambda$; (2) for all $B_j\in \mathcal{A},$ $\|B_j\|=k_j>\lambda$, and not all $k_j$ are equal. The only known examples of $\lambda$-designs are so called of type-1 designs, which are obtained from symmetric designs by a certain complementation procedure. Ryser and Woodall had independently conjectured that all $\lambda$-designs are of type-1. Suppose $r$ and $r^(r>r^)$ are replication numbers of $D$ and for distinct points $x$ and $y$ of $D$, let $\lambda(x,y)$ denote the number of blocks of $X$ containing $x$ and $y$. In this paper we investigate the possibilities of $\lambda$-designs to be of type-1 under the condition that $\|\lambda(x,y)-\lambda(x,y')\|< 2 \left(\dfrac{r-r^}{r+r^-2}\right)$. Under this condition, we prove that if $ \dfrac{r-1}{r^*-1} \le 3$, then $\lambda$-design $D$ is of type-1. Also we prove that $D$ has exactly two distinct block sizes.
کلیدواژه‌ها
$\lambda$-designs؛ Ryser-designs؛ Symmetric designs؛ $\lambda$-design conjecture؛ type-1 $\lambda$-designs

مراجع
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Some results on $\lambda$-design conjecture