Exact values of t(k) for k at least 5

Determine the exact value of t(k), the minimum alphabet size for which an infinite word avoids all tangrams with cut number at most k, for every integer k at least 5; in particular, determine t(5) within the known bounds 4 ≤ t(5) ≤ 6.

Background

The paper defines t(k) as the minimum alphabet size admitting an infinite word that avoids k-tangrams, where a k-tangram is a tangram with cut number at most k. It proves t(3) = t(4) = 4, resolving the previously stated question about t(3).

The construction used for 4-tangrams produces words that contain a 5-tangram, so it does not establish the corresponding result for k = 5. The authors explicitly state that the exact values of t(k) remain unknown for all k at least 5 and give the current bounds 4 ≤ t(5) ≤ 6. They also note that extending the approach to t(5) may be difficult because the pattern set S_5 is expected to be large.