401 and beyond: improved bounds and algorithms for the Ramsey algebra search

In this paper, we discuss an improvement of an algorithm to search for primes $p$ and coset-partitions of Z/pZ* that yield Ramsey algebras over Z/pZ. We also prove an upper bound on the modulus p in terms of the number of cosets. We have, as a corollary, that there is no prime $p$ for which there exists a partition of Z/pZ* into 13 cosets that yields a 13-color Ramsey algebra. Thus A263308(13) = 0.

[1]  ADDITIVE PROPERTIES OF MULTIPLICATIVE SUBGROUPS OF p , 2012 .

[2]  Jeremy F. Alm,et al.  Sum-free cyclic multi-bases and constructions of Ramsey algebras , 2015, Discret. Appl. Math..

[3]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[4]  R. Greenwood,et al.  Combinatorial Relations and Chromatic Graphs , 1955, Canadian Journal of Mathematics.

[5]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

[6]  Jeremy F. Alm,et al.  A fast coset-translation algorithm for computing the cycle structure of Comer relation algebras over Z/pZ , 2019, Theor. Comput. Sci..

[7]  B. Green.  Sum-product phenomena in F_p: a brief introduction , 2009, 0904.2075.

[8]  Jeremy F. Alm.  On sets of first-order formulas axiomatizing representable relation algebras , 2016, 1604.08227.

[9]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[10]  Roger D. Maddux,et al.  Relation Algebras , 1997, Relational Methods in Computer Science.

[11]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[12]  Representability of Ramsey Relation Algebras , 2015 .

[13]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[14]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[15]  R. Maddux.  Some varieties containing relation algebras , 1982 .

[16]  Earl Glen Whitehead.  Difference sets and sum-free sets in groups of order 16 , 1975, Discret. Math..

[17]  R. Graham,et al.  Quasi-random subsets of Z n , 1992 .