Colorings of hypergraphs with large number of colors

On the Erd{\H o}s--Hajnal problem in the case of 3-graphs

“…and in particular L 3 ≥ 4/e 3 > 0.199. Combining two previous arguments with Cherkashin-Kozik approach [4] Akolzin and Shabanov [1] proved that m(n, r) ≥ c n ln n r n , without explicit bounds on c. We show that this method gives the bound L 3 > 0.205 in Section 3. Cherkashin and Petrov [3] suggested an approach, based on the evaluation of the inverse function, to show that the sequence m(n, r)/r n has a limit.…”

“…So the case n = 3 is in some sense the most interest. Also note, that using the same inequality with better bounds on Turán numbers Akolzin and Shabanov [1] showed that m(n, r) < Cn 3 ln n • r n .…”

“…We rewrite the proof from [1] to get optimal constant in the case n = 3. First, for every vertex v introduce the weight w(v) as randomly (accordingly to the uniform distribution and independently) chosen number from [0, 1].…”

Equitable colorings of hypergraphs with few edges

“…In fact, the result of Kozik and Cherkashin remains true for large r compared to n, but in this case, Akolzin and Shabanov proved more stronger estimates [3]. Reader can find more results on the related problems in [4]- [7].…”

Regular behaviour of the maximal hypergraph chromatic number

“…So the case n = 3 is in some sense the most interest. Using the same inequality with better bounds on Turán numbers Akolzin and Shabanov [2] showed that m(n, r) < Cn 3 ln n • r n .…”