Asymptotic behavior of \(t\)uniform hypergraph Ramsey numbers
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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 \(t \geq 3\), \[ c \log_{t1} n < r_t(n,n) < c' \log _{t1} n \] where \(\log _u n \) denotes the \(u\)fold iterated logarithm and \(c\) and \(c'\) depend only on \(t\).
Bibliography  

1  P. Erdös, A. Hajnal and R. Rado. Partition relations for cardinal numbers, Acta Math. cad. Sci. Hungar. 16 (1965), 93196.
