# Rainbow matchings in k ‐partite hypergraphs

@article{Kiselev2020RainbowMI, title={Rainbow matchings in k ‐partite hypergraphs}, author={S. G. Kiselev and Andrey B. Kupavskii}, journal={Bulletin of the London Mathematical Society}, year={2020}, volume={53} }

In this paper, we prove a conjecture of Aharoni and Howard on the existence of rainbow (transversal) matchings in sufficiently large families F1,…,Fs of tuples in {1,…,n}k , provided s⩾470 .

## 6 Citations

Rainbow version of the Erd\H os Matching Conjecture via Concentration

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- 2021

We say that the families $\mathcal F_1,\ldots, \mathcal F_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,\ldots, F_{s+1}$, where $F_i\in…

Asymptotics of the Independence Number of a Random Subgraph of the Graph G(n, r, < s)

- MathematicsDoklady Mathematics
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Co-degree threshold for rainbow perfect matchings in uniform hypergraphs

- Mathematics
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Let k and n be two integers, with k ≥ 3, n ≡ 0 (mod k), and n sufficiently large. We determine the (k−1)-degree threshold for the existence of a rainbow perfect matchings in n-vertex k-uniform…

On the rainbow matching conjecture for 3-uniform hypergraphs

- MathematicsScience China Mathematics
- 2021

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the…

Rainbow perfect matchings for 4-uniform hypergraphs

- Mathematics
- 2021

Let $n$ be a sufficiently large integer with $n\equiv 0\pmod 4$ and let $F_i \subseteq{[n]\choose 4}$ where $i\in [n/4]$. We show that if each vertex of $F_i$ is contained in more than ${n-1\choose…

The Erd\H{o}s Matching Conjecture and concentration inequalities.

- Mathematics
- 2018

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint…

## References

SHOWING 1-10 OF 21 REFERENCES

Proof of the Erdős matching conjecture in a new range

- Mathematics
- 2017

AbstractLet s > k ≧ 2 be integers. It is shown that there is a positive real ε = ε(k) such that for all integers n satisfying (s + 1)k ≦ n < (s + 1)(k + ε) every k-graph on n vertices with no more…

The Erd\H{o}s Matching Conjecture and concentration inequalities.

- Mathematics
- 2018

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint…

The Size of a Hypergraph and its Matching Number

- MathematicsCombinatorics, Probability and Computing
- 2012

This paper verifies the conjecture that for any $t < \frac{n}{3k^2}$, every k-uniform hypergraph on n vertices without t disjoint edges has at most max $binom{kt-1}{k}-\binom-n-t-t+1-k$ edges.

A Rainbow r-Partite Version of the Erdős–Ko–Rado Theorem

- MathematicsCombinatorics, Probability and Computing
- 2017

Let [n] r be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k−1)n r−1 edges in [n] r contains a matching of size k. We…

Explicit construction of linear sized tolerant networks

- Mathematics, Computer ScienceDiscret. Math.
- 1988

Beyond the Erd\H{o}s Matching Conjecture

- Mathematics
- 2019

A generalization of the Erdős-Ko-Rado theorem is proved, which states that for $n> s^2k$ the largest family with property U(s,s(k-1)+1)$ is the star and is in particular intersecting.

On Rainbow Matchings for Hypergraphs

- MathematicsSIAM J. Discret. Math.
- 2018

It is shown that for sufficiently large $n$ with $\sum_{i=1}^t k_i\leq n(1-(4k\ln n/n)^{1/k}) $, the bound is tight.

Simple juntas for shifted families

- MathematicsArXiv
- 2019

Very general approximation by juntas results for shifted families with explicit (and essentially linear) dependency on the input parameters are presented.

A PROBLEM ON INDEPENDENT r-TUPLES

- Mathematics
- 1965

then G(n ; l) contains k independent edges . It is easy to see that the above result is best possible since the complete graph of 2k-1 vertices and the graph of vertices x1, . . ., xk-1 ; Yl, • • •,…

Concentration Inequalities and Martingale Inequalities: A Survey

- Mathematics, EconomicsInternet Math.
- 2006

We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general…