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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The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family Fnkthat does not contain 𝑠 pairwise disjoint sets is max. Ak,s,Bn,k,s, where Ak,s=sk1k and Bn,k,s=Bnk:Bs1. 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 sk1k=s1sskk with respect to minimal degree. As a consequence we derive limkfsk+1,k,ssk+1k<s1sεsfor some positive constant 𝜀𝑠 which depends only on 𝑠.

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Studia Scientiarum Mathematicarum Hungarica
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ISSN 0081-6906 (Print)
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