Ramsey Theory and Geometry of Closed Loops

Parts A and B of the paper are consistent with the standard R(3,3)=6 argument for slope–sign colorings on K6 and with the Jordan-curve same/different-region coloring examples (including the case where green triangles are impossible and red ones must appear) . However, the paper repeatedly asserts R_trans,intrans(2,3)=3 for the two-class same/different (transitive green, intransitive red) setting, which is false; 3 (and even 4) vertices do not force a monochromatic triangle, while 5 do by pigeonhole, exactly as the model argues . The paper also gives an inadequate justification for the nonzero-slope assumption (α_ik≠0) by appealing to intersections with y=0, which does not ensure pairwise distinct y-coordinates for all chosen points, though this flaw does not affect the correctness of Part A’s main conclusion .