Erd\H osKoRado type results for partitions via spread approximations
Abstract
In this paper, we address several Erd\H osKoRado type questions for families of partitions. Two partitions of $[n]$ are {\it $t$intersecting} if they share at least $t$ parts, and are {\it partially $t$intersecting} if some of their parts intersect in at least $t$ elements. The question of what is the largest family of pairwise $t$intersecting partitions was studied for several classes of partitions: Peter Erd\H os and Székely studied partitions of $[n]$ into $\ell$ parts of unrestricted size; Ku and Renshaw studied unrestricted partitions of $[n]$; Meagher and Moura, and then Godsil and Meagher studied partitions into $\ell$ parts of equal size. We improve and generalize the results proved by these authors. Meagher and Moura, following the work of Erd\H os and Székely, introduced the notion of partially $t$intersecting partitions, and conjectured, what should be the largest partially $t$intersecting family of partitions into $\ell$ parts of equal size $k$. The main result of this paper is the proof of their conjecture for all $t, k$, provided $\ell$ is sufficiently large. All our results are applications of the spread approximation technique, introduced by Zakharov and the author. In order to use it, we need to refine some of the theorems from the original paper. As a byproduct, this makes the present paper a selfcontained presentation of the spread approximation technique for $t$intersecting problems.
 Publication:

arXiv eprints
 Pub Date:
 August 2023
 DOI:
 10.48550/arXiv.2309.00097
 arXiv:
 arXiv:2309.00097
 Bibcode:
 2023arXiv230900097K
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics