Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Earlier approaches to the seven-cycle color bound
c7_adaptive_frames: Obstructions, signed and positive repairs, and explicit lower bounds for the adaptive frame operator.
c7_core_port_constructions: An inequality excluding a broad family of clique cores connected through bipartite ports as threshold counterexamples.
c7_correlated_ports: A squared-degree color bound handles arbitrary correlated attachments and arbitrary cores completely joined to independent hubs.
c7_cut_palette_certificate: An enlarged cut dual proves the savings bound for a triangular maximum star, an independent nonneighborhood, or a saturated complementary cut.
c7_cycle_space_obstruction: Reasons that a direct binary cycle-space or color-map rank argument does not establish the C7 color bound.
c7_dense_curve_obstruction: A proved three-branch construction disproves the C7 extension of the full Bucić–Chen–Ma curve, but not the threshold conjecture.
c7_dominating_walk_cliques: A joint two- and three-walk clique dominating outside degrees proves the savings bound; a heavy-degree corollary and selection obstructions.
c7_half_edge_reduction: An equivalent half-edge target and a physical-rectangle route, with a proved Ore-heavy clique and the remaining localization gap.
c7_large_palettes: Independent active palettes can have arbitrary fixed size above the Turan density with minimum degree arbitrarily close to one half.
c7_linked_tripartite_core: A degree-weighted three-walk clique proves the squared-density color bound for arbitrary tripartite triangle networks attached to a properly three-colored triangle-free core.
c7_local_rainbow_sets: Rainbow sets arising from robust three-edge paths or from a fixed triangle; neither supplies a full threshold reduction.
c7_ore_mass_obstruction: A super-Turan family has Ore-heavy endpoints on only four ninths of its vertices, ruling out a canonical mass-localization shortcut.
c7_overlapping_ports: A sharp resource inequality excluding clique cores with overlapping independent neighborhoods in a triangle-free port graph.
c7_palette_stationarity: LP duality and weight stationarity for density maximization expose why ordinary vertex symmetrization has not closed the palette gap.
c7_private_core_hubs: A proved density bound for private clique cores attached to the vertices of an arbitrary triangle-free hub graph.
c7_rainbow_representatives: A super-Turan construction rules out even the square-root edge-count spectral bound for rainbow representative subgraphs.
c7_random_blowup_lp: Fractional-coloring formulas for complete and random finite blow-ups, and an induced-matching construction principle.
c7_reflection_norms: Checked limitations of direct reflection-folding, graph-norm, and polynomial spectral-kernel approaches to the C7 color bound.
c7_residual_geometry: Optimal-dual neighborhood inequalities and conditional density bounds narrow the savings problem, but their uncolored relaxation is false. A full-capacity template also obstructs unconditional pair balancing.
c7_residual_spectral: An exact spectral minimax formula for color savings, concavity in squared vertex weights, and a universal degree-deficit spectral bound.
c7_second_degree: A second-degree bound and the unresolved three-path clique enlargement approach.
c7_semidefinite_approach: An algebraic approach through positive semidefinite edge kernels; certificates, trace obstructions, and the unresolved global step.
c7_singleton_palettes: A compatible-pair degree inequality forces singleton palettes above the Turan threshold and gives a density-budget identity there.
c7_tensor_exclusions: Analytic obstructions exclude private-core tensors with product weights and every power of a looped three-vertex path with arbitrary weights.
c7_tensor_palettes: Categorical products can reuse both edge orientations for nontriangle palettes, giving a proved amplification criterion but no counterexample.
c7_triangle_average: A universal average bound for density at least five sixteenths, and a reduction leaving at most five nontriangular types.
c7_triangle_component: A sharp weighted partition bound localizes the triangle-vertex mass inequality to one component of the triangular-edge graph.
c7_triangle_vertices: A weighted symmetrization lemma for super-Turan graphs and its limited relevance to the C7 color problem.
c7_triangular_support: A two-star rectangle proves the squared-density bound when every supported edge is triangular, including arbitrary thinned demands.
c7_two_nontriangular_types: An average inequality for two adjacent nontriangular types, and resulting rectangle bounds when all triangular vertices form a three-walk clique.
c7_two_star_rectangles: Anchor optimization, a two-star family of three-walk cliques, and an unconditional high-density color bound; threshold localization remains open.
c7_universal_components: The half-edge inequality holds when each triangular-edge component has universal two- and three-walk connectivity, via a deficit inequality.
c7_universal_subsets: Palette and replacement inequalities for a short-walk universal subset identify the attachment-triangle obstruction beyond isolated components.
c7_walk_clique_pruning: A maximum-degree separation inequality and a twin-box induction prove the sharp mass bound for an odd-three-walk clique; color localization and simultaneous two-walk connectivity remain unresolved.
evidence/: An exact symbolic certificate for the two-type identities and bounded exploratory palette searches supporting archived research notes.
These pages retain proved restricted results, counterexamples to stronger assertions, finite searches, and approaches that did not establish the threshold. Their local open questions are historical: the author-recorded C7 proof gives an argument for the threshold, and the Lean account records the Lean development behind claim L17, accepted at tier 2 on 2026-09-25 for the Lean sources and the statement as they stood on 2026-09-25T03:40:15Z and first carried by the default branch on 2026-09-28.
For context, first read the random-blow-up palette model and half-edge reduction. The dense-curve construction shows why the stronger Bucić–Chen–Ma full-density formula cannot be extended unchanged to . The reflection-norm note records the limitations checked for that route. The evidence guide locates the exact symbolic certificate and the bounded searches cited by the archive pages.