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A short proof of Erdős' conjecture for triple systems.
- Source :
-
Acta Mathematica Hungarica . Apr2017, Vol. 151 Issue 2, p495-509. 15p. 1 Diagram. - Publication Year :
- 2017
-
Abstract
- Erdős [1] conjectured that for all $${k \geq 2}$$ , $${s \geq 1}$$ and $${n \geq {k(s+1)}}$$ , an n-vertex k-uniform hypergraph $${\mathcal{F}}$$ with $${\nu(\mathcal{F})=s}$$ cannot have more than $${\max\{\binom{sk+k-1}k,\binom nk-\binom{n-s}k\}}$$ edges. It took almost fifty years to prove it for triple systems. In [5] we proved the conjecture for all s and all $${n \geq 4(s+1)}$$ . Then Łuczak and Mieczkowska [6] proved the conjecture for sufficiently large s and all n. Soon after, Frankl proved it for all s. Here we present a simpler version of that proof which yields Erdős' conjecture for $${s \geq 33}$$ . Our motivation is to lay down foundations for a possible proof in the much harder case k = 4, at least for large s. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 02365294
- Volume :
- 151
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Acta Mathematica Hungarica
- Publication Type :
- Academic Journal
- Accession number :
- 121626391
- Full Text :
- https://doi.org/10.1007/s10474-017-0692-8