On Colorings of Random Hypergraphs with Large Uniformity

Student: Davletshin Farid

Educational Programme: Mathematics (Bachelor)

Year of Graduation: 2021

In 1964 P. Erdős proved, that k-graph (i. e. k-uniform hypergraph) with \dfrac{k^2}{2} vertices and m(k)=(1+o(1))\dfrac{e \ln2}{4}k^22^k randomly chosen edges of size k is asymptotically almost surely is not two colorable [1]. D. Shabanov, J. Kozik and L. Duraj (2020) showed that m(k) is \textit{sharp thresold} [2]. This means that for any \epsilon > 0, any k-graph with (1-\epsilon)m(k) randomly and uniformly chosen edges is asymtotically almost surely two colorable. This paper is adaptation of the second moment method which was used in the research above. We prove analogous theorem as in [2] but for three colors. We are considering equitable colorings.

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