Asymptotic behavior of \(t\)-uniform hypergraph Ramsey numbers

A \( t\)-graph has a vertex set \( V\) and an edge set \( E\) consisting of some prescribed set of \( t\)-subsets of \( V\). For \( t\)-graphs \( G_i\), \( i=1,\dots,k\), let \( r_t(G_1,\dots,G_k)\) denote the smallest integer \( m\) satisfying the property that if the edges of the complete \( t\)-graph on \( m\) vertices are colored in \( k\) colors, then for some \( i\), \( 1\leq i\leq k\), there is a subgraph isomorphic to \( G_i\) with all \( t\)-edges in the \( i\)-th color. We denote \( r_t(n_1, \dots, n_k)= r_t(K_{n_1}, \dots, K_{n_k})\).

Conjecture (proposed by Erdös, Hajnal and Rado [1])

For every \(t \geq 3\), \[ c \log_{t-1} n < r_t(n,n) < c' \log _{t-1} n \] where \(\log _u n \) denotes the \(u\)-fold iterated logarithm and \(c\) and \(c'\) depend only on \(t\).

1 P. Erdös, A. Hajnal and R. Rado. Partition relations for cardinal numbers, Acta Math. cad. Sci. Hungar. 16 (1965), 93-196.