Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 62: Issue 1
A Note on the Erdős Matching Conjecture
Authors:
Ryan R. Martin Iowa State University, Ames, Iowa, USA

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Balázs Patkós HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

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Pages:
63–69
Online Publication Date:
25 Mar 2025
Publication Date:
09 Apr 2025
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2025.04330
Keywords:
Extremal set theory; matchings; stability
Open access

The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family F⊆nkthat does not contain 𝑠 pairwise disjoint sets is max. Ak,s,Bn,k,s, where Ak,s=sk−1k and Bn,k,s=B∈nk:B∩s−1≠∅. The case 𝑠 = 2 is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case 𝑛 = 𝑠𝑘 was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if 𝑘, 𝑠 are fixed and 𝑛 is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for 𝑛 ∈ [𝑠𝑘, 𝑠𝑘 + 𝑐𝑠,𝑘 ] if 𝑠 is large enough compared to 𝑘. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most 𝑠 in a similar range of parameters.

In this short note, we are concerned with the case 𝑠 ≥ 3 fixed, 𝑘 tending to infinity and 𝑛 ∈ {𝑠𝑘, 𝑠𝑘 + 1}. For 𝑛 = 𝑠𝑘, we show the stability of the unique extremal construction of size sk−1k=s−1sskk with respect to minimal degree. As a consequence we derive limk→∞fsk+1,k,ssk+1k<s−1s−εsfor some positive constant 𝜀𝑠 which depends only on 𝑠.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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