Lower bounds for hypergraph Ramsey numbers
For \(m_1, m_2,\dots, m_k\) with \(m_i\geq r\), \(R(m_1,\dots, m_k; r)\) denotes the least integer \(n\) such that when \(f\) is any \(k\)- coloring of the \(r\)-subsets of an \(n\)-set there is an index \(i\) and an \(m_i\) subset such that all of its \(r\)-subsets are assigned the \(i\)th color. If \(m_1= m_2=\dots = m_k= m\), we write \(R_k(m; r)\) for \(R(m_1,\dots, m_k; r)\). The main result of this paper is: For \(k\geq 2\), \(h\geq 2\), \(m\geq r\), \(r\geq 3\) and \(2\leq q< r\), we have \(R(m, R_k(r; q),(r- 1)^h+ 1; r)> (R_k(m; q)- 1)^h\).
scientific article; zbMATH DE number 790498