# On the packing for triples

## TL;DR

This paper investigates bounds on the maximum size of 3-packings in combinatorial design theory, providing exact, near-exact, and linear bounds for large n relative to k.

## Contribution

It establishes new upper and lower bounds for 3-packings, including cases with exact and near-exact bounds, advancing understanding of packing limits in combinatorics.

## Key findings

- Exact bounds for certain 3-packings
- Bounds differ by a constant in some cases
- Bounds differ linearly in n in one case

## Abstract

For positive integers $n\geq k\geq t$, a collection $ \mathcal{B} $ of $k$-subsets of an $n$-set $ X $ is called a $t$-packing if every $t$-subset of $ X $ appears in at most one set in $\mathcal{B}$. In this paper, we give some upper and lower bounds for the maximum size of $3$-packings when $n$ is sufficiently larger than $k$. In one case, the upper and lower bounds are equal, in some cases, they differ by at most an additive constant depending only on $k$ and in one case they differ by a linear bound in $ n $.