Density Hales-Jewett and Moser numbers

A note on multiparty communication complexity and the Hales–Jewett theorem

“…Theorem 10 actually proves a lower bound on the maximal size of a Fujimura set in ∆ n,k , similarly to the proof in [19]. In fact…”

“…Thus, the bound in [7] on P art n,k using the Hales-Jewett theorem is in fact tight, and the density Hales-Jewett number can be thought of as a quantity in communication complexity. This gives a new angle to this well studied quantity.As a simple application we prove a lower bound on c n,k , similar to the lower bound in [19] which is roughly c n,k /k n ≥ exp(−O(log n) 1/⌈log 2 k⌉ ). This lower bound follows from a protocol for P art n,k .…”

“…As a simple application we prove a lower bound on c n,k , similar to the lower bound in [19] which is roughly c n,k /k n ≥ exp(−O(log n) 1/⌈log 2 k⌉ ). This lower bound follows from a protocol for P art n,k .…”

“…The proof of [18] is the first combinatorial proof of the density Hales-Jewett theorem, and also provides effective bounds for c n,k . In a second paper [19] in this project, several values of c n,3 are computed for small values of n. Using ideas from recent work [9,13,17] on the construction of Behrend [4] and Rankin [21], they also prove the following asymptotic bound on c n,k . Let r k (n) be the maximal size of a subset of [n] without an arithmetic progression of length k, then:…”

“…We prove the relationship between c n,k and α(P art n,k ) in Section 2, and the lower bound on c n,k is proved in Section 3. Lastly, Section 4 contains a discussion on Fujimura sets, mentioned in [19], and their communication complexity analogues.…”