Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 16

Approach registry

The common target is the finite theorem (F) in the main argument: finite configurations with one hereditary extraction constant and unbounded actual dissociated coloring number. The research is open. Extraction is established for the prime tensors, the linear-side line covers and matchings, the critical Boolean moments, and the fixed- $C$ simplicial boundaries; no family has a proved unbounded actual coloring number. Every row below is author-recorded. A route is closed only where its page states a proved obstruction; a closed row names it, and the page states what would reopen the route.

Live targets

Three families have uniform extraction and a precise surviving obligation. Their priority rests on the mathematical reasons given.

Route	Exact obligation	Standing
Simplicial boundaries at one fixed $C$	With $k=\lfloor n/(C\log n)\rfloor$ , force a nonzero monochromatic ternary cycle under every fixed finite coloring	Uniform extraction is proved. Three colors avoid every decomposable cycle and two colors avoid all joined boundaries and small-vertex-excess cycles, so a proof needs a color-adaptive nondecomposable cycle mechanism; rational homology does not supply it. Highest priority: the extraction proof is exact and the surviving obligation is the most sharply isolated.
Ordinary incidence covers and matchings	Coordinate the varying relation supports of line covers, or of the enlarged direction-matching family, to defeat every finite palette	Extraction holds for all covers and all matchings; fixed-voltage automata expose every ternary relation but give no density lower bound; canonical single-circuit lifts have two colors and fixed templates are three-splittable, so an amplifier must couple different relation spaces; bounded periods are excluded without a comparison against the automaton's periods.
Critical Boolean moments	Produce a nonzero monochromatic ternary function with the required vanishing moments under every finite palette	Extraction is proved through semistability and the weak stable-lattice bound; palette-dependent height is exponential in the palette, not one; full tensors lose extraction and pointwise ternarization has discrete obstructions, so sparse or nonpointwise trades are the open direction.

Routes with a proved extraction side and an open coloring side

Route	Exact obligation	Standing
Distinct-prime tensors	Prove unbounded dissociated coloring for the grids over all odd primes, equation (4) of the main argument	Uniform extraction proved for all prime grids and all roots of unity; ordinary coloring equals three through prime 19 and lies in $\{3,4\}$ at 23 (reset and plateau); arc colorings bound every grid by the least flat cyclotomic degree, so (4) requires signed vanishing sums in arbitrarily short arcs, the negation of the open claim L124, whose proof would close this route; coloring unresolved
Linear-side axis lines	Prove unbounded dissociated coloring of $L(3,4,\ldots,t+2)$	Direct uniform extraction proved; the actual-color trade problem remains; full tensor products lose uniform extraction
Concrete field, flag and polynomial lines	Force arbitrary-color mixed trades	Uniform extraction is proved; every binary affine-offset checksum fails at stage two, but arbitrary two-coloring and unbounded actual coloring remain open
Quotient point relations	Obtain color-safe preimages using joint information beyond individual class spans	Quotient coloring is unbounded; existential colored lifting holds in two directions but fails in three
Small-loss difference towers	Obtain arbitrary-color growth on varying line seeds while keeping a positive product retention	Near-lossless extraction and exact height halving proved; valuation-separated seeds stay bounded-color and canonical lifts split
Fixed-height adaptive focusing	Prove infinite coloring number at one fixed finite coefficient height	A complete focusing reduction is proved; the available palette-dependent height does not supply its hypothesis; the piecewise-syndetic reformulation is an equivalent target, and ordinary recurrence with infinite rational coloring number does not suffice
Bounded height and determinant flattening	Force a monochromatic Hall-tight determinant support with controlled coefficient magnitudes	Uniform height and conditional flattening proved; pattern-palette and coprime-conductor focusing losses unresolved
Variable-height coefficient growth	Force monochromatic factors satisfying the weighted-simplex counting inequality	Coefficient-safe flattening beyond Hall is proved; profile focusing has no compatible growth bound, and actual root circuits escape doubly exponentially
Coefficient-safe profile compression	Derive enough coordinate-profile merging from dissociation, or use its block-sum mechanism more generally	Summable reciprocal profile-deficit theorem; bounded-state colorings excluded, arbitrary distinct profiles not excluded
Same-conductor Galois squaring	Control collisions and actual colors of pair products in the coefficient-two squaring identity	Identity and nontriviality proved; surviving coefficients need not be ternary or monochromatic
Cyclotomic binary lifting	Turn arithmetic circuit structure into unrestricted, actual-color descent	Exact binary threshold nineteen and singleton descent proved; a finite-support theorem is not the fixed-height premise
Tensor threshold amplification	Restart a small signed-span omission invariant after each chromatic jump	Cheap amplification proved; maximal root-coverage reset at the three-prime jump; upper separation lifts
Root transversals	Exclude dissociated prime-fiber transversals for arbitrarily large primes, without a symmetry assumption	Exact finite-coloring equivalence; the $2^k$ bound covers only equivariant partitions

Closed routes and their obstructions

Route	What was sought	Proved obstruction
Density-only descent in prime grids	Fiber-free one-prime descent from high short-relation rank	The saturated prime staircase has connected integral height-two circuit rank of density tending to one yet mandatory coordinates and only full-fiber next-prime supports; two-factor products of its circuits have two colors; the pivot family at density $1/20$ and the nonintegral equal-modulus example are the weaker precursors. A new route must couple actual colors, not raise a rank density.
Small-loss and Hadamard anchor gluing	A reusable next-color anchor from a fixed source	The Grow source is excluded at every arity by the large-arity obstruction and the boundary certificate; quadratic analytic gluing does not propagate anchors; mobile buffers above a source-dependent length prevent increase; Hadamard feedback loses every anchor at large order and under diagonal padding. Reopening needs a different source with a surviving anchor.
Ramsey and interval systems	Leave bounded-support representations while controlling the full signed closure	The fixed-width route is ruled out, including the actual fixed-order Nešetřil–Rödl–Sales family; fixed-alphabet parameter words admit a coloring with rationally independent monochromatic images. Increasing alphabets or non-coordinatewise encodings need a new relation-preservation argument.
Rational relation spaces	A structured signed kernel with both required properties	Exact integer realization proved; an explicit saturated lattice shows raw span dimension is not a viable surrogate
Independently planted relations	Structured orientations that avoid label collapse	Independent-sign obstruction proved; no live random candidate
Prescribed relation menus and private lifts	Color forcing from prescribed determinant supports or relation-preserving lifts	Every prescribed menu is two-splittable even with same-color anchored axes; private-prime lifts preserve all integer relations yet always admit two rationally independent colors. Adaptive relation selection is not excluded.
Fixed disjoint-box complexity	A relation family whose bounded box complexity forces colors	Every fixed number of disjoint constant-sign boxes is finitely colorable, 16 colors for two boxes, and chains needing $k$ boxes admit one 30-coloring; box complexity must grow and be coupled to colors.
Polynomial fibers as seed amplifiers	Extraction-preserving flat representations of seed relations	Prouhet identities flatten on a focused progression, but that fiber already has its own relation and full fibers lose extraction
Full-shift prescribed supports	Unbounded coloring from long-shift relation hypergraphs	Uniform extraction bounds long-shift prescribed coloring by $O_\delta(k\log k)$ , and the shift-hypergraph bound gives six colors in the high-fractional regime
Fixed-template and cofactor forcing in covers	A fixed connected repeated-type template, or a cofactor template family, unavoidable under all colors	Fixed repeated-type templates split with three colors; rank-correct cofactor templates on three-universal-row bases split with seven; signed-cographic gluing has at most three colors; the interval amplifier loses extraction as it grows. Varying templates under arbitrary colors remain the target.
Affine and checksum palettes	Exclusion of all finite palettes through algebraic palette classes	Generic voltages exclude affine homomorphic palettes at rational density and finite-abelian checksums fail at the moment boundary; arbitrary palettes are not algebraic

The scale-separated block union is a proved reduction, not another candidate construction: ordinary direct sums alone cannot amplify the coloring number, since the coloring number of a separated union is the maximum of its blocks' coloring numbers. Finite tensor checks show that a larger side size can decrease the coloring number, so monotone growth in the side sizes must not be assumed. A vanishing rational rank fraction, a fixed $h$ -freeness, or a decaying extraction constant is not the target.

Linked from (1)

Dissociated extraction and coloring

Graph