Ramsey theory is a central and active field of combinatorics. Although his Ramsey numbers for graphs have been extensively investigated since his Ramsey work in the 1930s, there is still an exponential gap between the best known lower and upper bounds. For the $k$-uniform hypergraph, the bounds are tower-shaped and the height increases with $k$ . We now generalize Ramsey’s theorem multidimensionally to the Cartesian product of graphs to prove that double-exponential upper bounds are sufficient in all dimensions. More precisely, for all positive integers $r,n,d$, any $r$ In the color scheme, if , $N\geq 2^{2^{C_drn^{d}}}$ , there is a copy of $\square^{d} K_n$ such that the edges in each direction are solid. As an application of our approach, we also get an improvement on the multidimensional Erd\H{o}s-Szekeres theorem proved by Fishburn and Graham $30$ years ago. Their boundaries were recently improved by Buci\’c, Sudakov, and Tran. They gave an upper bound that is an exponential function of 3 times over 4 dimensions. Improve the result showing that the double exponential upper bound preserves any number of dimensions.

    Source link


    Leave A Reply