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arXiv · 2108.07692

An Extension of the Erd\H{o}s-Ko-Rado Theorem to uniform set partitions

Karen Meagher·Mahsa N. Shirazi·Brett Stevens
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Abstract

A $(k,\ell)$-partition is a set partition which has $\ell$ blocks each of size $k$. Two uniform set partitions $P$ and $Q$ are said to be partially $t$-intersecting if there exist blocks $P_{i}$ in $P$ and $Q_{j}$ in $Q$ such that $\left| P_{i} \cap Q_{j} \right|\geq t$. In this paper we prove a version of the Erd\H{o}s-Ko-Rado theorem for partially $2$-intersecting $(k,\ell)$-partitions. In particular, we show for $\ell$ sufficiently large, the set of all $(k,\ell)$-partitions in which a block contains a fixed pair is the largest set of 2-partially intersecting $(k,\ell)$-partitions. For for $k=3$, we show this result holds for all $\ell$.

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Karen Meagher, Mahsa N. Shirazi, Brett Stevens. 2021-08-17. An Extension of the Erd\H{o}s-Ko-Rado Theorem to uniform set partitions. https://arxiv.org/abs/2108.07692

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