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• W4295127209 abstract "Let $ H = (V,E) $ be a hypergraph. By the chromatic number of a hypergraph $ H = (V,E) $ we mean the minimum number $chi(H)$ of colors needed to paint all the vertices in $ V $ so that any edge $ e in E $ contains at least two vertices of some different colors. Finally, a hypergraph is said to form a clique, if its edges are pairwise intersecting. In 1973 ErdH{o}s and Lov'asz noticed that if an $n$-uniform hypergraph $ H = (V,E) $ forms a clique, then $ chi(H) in {2,3} $. They untoduced following quantity. $$ M(n) = max {|E|: exists {rm an} n-{rm uniform} {rm clique} H = (V,E) {rm with} chi(H) = 3}. $$ Obviously such definition has no sense in the case of $ chi(H) = 2 $. Theorem 1 (P. Erdos, L. Lovasz} The inequalities hold $$ n!(e-1) le M(n) le n^n. $$ Almost nothing better has been done during the last 35 years. At the same time, another quantity $ r(n) $ was introduced by Lovasz r(n) = max {|E|: ~ exists {rm an} ~ n-{rm uniform} ~ {rm clique} ~ H = (V,E) ~ {rm s.t.} ~ tau(H) = n}, $$ where $ tau(H) $ is the {it covering number} of $ H $, i.e., $$ tau(H) = min {|f|: ~ f subset V, ~ forall ~ e in E ~ f cap e neq emptyset}. $$ Clearly, for any $n$-uniform clique $ H $, we have $ tau(H) le n $, and if $ chi(H) = 3 $, then $ tau(H) = n $. Thus, $ M(n) le r(n) $. Lov'asz noticed that for $ r(n) $ the same estimates as in Theorem 1 apply and conjectured that the lower estimate is best possible. In 1996 P. Frankl, K. Ota, and N. Tokushige disproved this conjecture and showed that $ r(n) ge (frac{n}{2})^{n-1} $. We discovered a new upper bound for the r(n) (so for M(n) too). Theorem 2. $$ M(n) leq r(n) le c n^{n-1/2} ln n. $$, where c is a constant." @default.
• W4295127209 created "2022-09-11" @default.
• W4295127209 creator A5000434309 @default.
• W4295127209 date "2011-07-10" @default.
• W4295127209 modified "2023-09-27" @default.
• W4295127209 title "About maximal number of edges in hypergraph-clique with chromatic number 3" @default.
• W4295127209 doi "https://doi.org/10.48550/arxiv.1107.1869" @default.
• W4295127209 hasPublicationYear "2011" @default.
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