Geodesic
On Free Semigroups and Ramsey Numbers
Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 229-232

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If the length of a word w in a free semigroup F(X) satisfies , then for every partition of F(X) into k classes, w has n consecutive factors of length ≥p in the same class. As a consequence, the diagonal Ramsey numbers R(pn+1, p+1, k) have 1+pnk as lower bound. 
Lallement, Gerard. On Free Semigroups and Ramsey Numbers. Canadian mathematical bulletin, Tome 17 (1974) no. 2, pp. 229-232. doi: 10.4153/CMB-1974-045-5
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     author = {Lallement, Gerard},
     title = {On {Free} {Semigroups} and {Ramsey} {Numbers}},
     journal = {Canadian mathematical bulletin},
     pages = {229--232},
     year = {1974},
     volume = {17},
     number = {2},
     doi = {10.4153/CMB-1974-045-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1974-045-5/}
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