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A294676 Ramsey-Comer numbers: a(n) is the smallest prime p congruent to 1 mod 2n such that for every prime q >= p (also congruent to 1 mod 2n), the multiplicative subgroup H of (Z/qZ)* of index n contains a solution to x+y = z.

%I A294676 #13 Nov 17 2017 01:03:35
%S A294676 3,13,19,73,131,313,547,193,613,1201,1453,1249,547,2857,2971,1601,
%T A294676 4217,3169,2243,4441,9661,10957,7039,7873,8951,11701,14419,18257,
%U A294676 11311,29641
%N A294676 Ramsey-Comer numbers: a(n) is the smallest prime p congruent to 1 mod 2n such that for every prime q >= p (also congruent to 1 mod 2n), the multiplicative subgroup H of (Z/qZ)* of index n contains a solution to x+y = z.
%C A294676 a(n) <= n^4 + 5 (cf. Alm, 2017).
%C A294676 The subgroup H, along with its n-1 cosets, induces a cyclic coloring on K_q. Labeling the vertices 0 through q-1, color the edge uv by the color corresponding to the coset containing u-v (mod q). Thus if q >= a(n), the coloring induced by H and its cosets must contain a monochromatic triangle. In fact, it contains many monochromatic triangles in each color class.
%C A294676 The data gathered thus far suggest that the bound n^4 + 5 can be replaced by cn^3 for some c > 1, but there is no proof.
%C A294676 a(n) > A263308(n). The reason A263308(8) is zero can be taken to be that a(8) is exceptionally small; similarly, a(13) is small, so A263308(13)=0.
%H A294676 Jeremy F. Alm, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Alm/alm.html">401 and beyond: improved bounds and algorithms for the Ramsey algebra search</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.4. (Also here: <a href="https://arxiv.org/abs/1609.01817">arXiv:1609.01817</a>  [math.NT], 2016.)
%Y A294676 Cf. A263308.
%K A294676 nonn,more
%O A294676 1,1
%A A294676 _Jeremy F. Alm_, Nov 06 2017