Problemi aperti e metodi verificati

matematicaquantumproblemi apertimetodo di ricerca

272 pagine di matematica della ricerca: i problemi di Erdős, il problema PPT², i metodi che hanno funzionato e le rotte già esplorate e chiuse. Ogni pagina ha provenance e verifica; i pesi d'uso vengono dalla campagna reale.

• Additively indecomposable ordinals (Cantor's γ-numbers) — the ω^β / Cantor-normal-form-monomial ordinals
• Unbounded-degree number-field grids via infinite class-field towers (Golod–Shafarevich + point-counting)
• AlphaEvolve — LLM-guided evolutionary search over verifier-checked numeric parameter spaces
• B_h sets — generalized Sidon sets of order h (distinct h-fold sums)
• B_2[g] sequences — bounded (but not unique) representation, the Sidon relaxation
• Bisociation (Koestler 1964) — creative insight as the collision of two habitually unrelated frames of thought usata 12×
• Bound entanglement — entangled states that are PPT, hence undistillable; the rare structured regime where a PPT² counterexample must live usata 33×
• Cantor normal form — decomposition of every ordinal into a strictly-decreasing sum of ω^β monomials
• Choi–Jamiołkowski isomorphism — state–channel duality: a map is CP iff its Choi matrix is PSD usata 18×
• Chromatic-defect random extraction + sparse-core bootstrapping (the Erdős–Hajnal high-girth subgraph technique)
• Non-concentration of the chromatic number of a random graph (Heckel / Heckel–Riordan / Heckel–Panagiotou)
• Closed ordinal Ramsey numbers $R^{cl}(\\alpha,3)$ — the topological (closed-in-supremum) variant of ordinal partition calculus
• Complement-graph symmetry argument — linking two graph parameters via G and its complement Ḡ
• Covariance / symmetry reduction of quantum channels — THE WALL: the one recipe every partial PPT² result reuses, and nobody has escaped to the general case usata 16×
• Covering systems of congruences
• Erdős–Szekeres cup-cap (cap-cup) theorem: exact Ramsey number for convex chains
• Cycle-length spectrum C(G) of a graph as the core combinatorial object
• De Bruijn–Erdős compactness theorem — infinite chromatic number is determined by finite subgraphs
• r-degeneracy — bounded induced-subgraph minimum degree (Lick–White; coloring number, Erdős–Hajnal)
• Δ-system / sunflower — the core combinatorial object
• Dependent random choice — pick a small random test-tuple, take its common neighborhood; the resulting set is large and almost every small subset of it still has a large common neighborhood, giving a workhorse for embedding sparse/bipartite graphs into dense hosts
• Dilworth's theorem (chain/antichain decomposition of a poset) and its dual, Mirsky's theorem
• Discharging method — charge-counting technique for planar/structural graph coloring and cycle-existence proofs
• Discrete-logarithm explicit Sidon construction (Cilleruelo)
• Distortion method (Hough; Balister–Bollobás–Morris–Sahasrabudhe–Tiba) for covering-system impossibility bounds
• Dual-oracle verification — certify every claimed object by two independent oracles, count-certify exhaustive runs, and scan all prior-art branches before claiming novelty usata 45×
• Duffin–Schaeffer conjecture / metric Diophantine approximation (Koukoulopoulos–Maynard theorem)
• Block-decomposition + weighted Cauchy–Schwarz energy method (Erdős–Stöhr–Halberstam–Roth–O'Bryant)
• Minimum edge-deletion to bipartite: β(G)/h_G(n), the Max-Cut complement, and the Edwards–Erdős bound
• Elekes–Sharir(–Guth–Katz) reduction: distinct distances → point-line incidences in SE(2)
• Entanglement-breaking channel — measure-and-prepare maps: separable Choi matrix ⇔ destroys all entanglement usata 12×
• Entanglement witness — a Hermitian W with Tr(Wσ)≥0 on all separable states but Tr(Wρ)<0 on some entangled ρ usata 6×
• Entropy method — Shannon-entropy / coding-theoretic proof technique (Rao, Tao)
• Entropy-inequality method (Katz–Tardos): Shannon-entropy linear programming for the sums-and-entries / distinct-distances problem
• The alteration (deletion) method — probabilistic existence proofs that build an almost-good random structure, then delete its blemishes; canonical instance: Erdős's 1959 high-girth/high-chromatic-number graphs
• Erdős–Fuchs theorem: average representation count can't be too close to linear
• Erdős–Hajnal conjecture: forbidding one induced subgraph forces polynomial-size cliques/independent sets
• Erdős–Kleitman matching problem — 2026 resolution of e(n,s)
• Erdős–Rankin construction (covering-congruences translation for large prime gaps)
• Erdős–Rényi random-subset-plus-deletion construction (power-law random sets for additive bases)
• Random-set construction + Janson's inequality / Vu concentration (the Poisson paradigm)
• Finite-field / projective-plane constructions for extremal additive sets
• Finite-field parabola/Sidon-block construction with lacunary-scale gluing
• Flag algebras — Razborov's SDP-based calculus for extremal graph/hypergraph densities
• Flecnode polynomial / ruled-surface geometry (G. Salmon)
• GCD graphs (Koukoulopoulos–Maynard graph-theoretic sieve)
• Generalized Sidon set constructions — the $B_h[g]$ toolbox (finite-field unions, interleaving, products)
• $\gamma$-Golomb rulers (bounded-multiplicity difference sets generalizing Sidon sets)
• Greedy pointwise-extension argument exploiting shared interior points of APs
• Harris–FKG correlation inequality for increasing/decreasing events (random graphs, percolation)
• Pippenger–Spencer hypergraph covering theorem (and the FGKMT generalization)
• Hypergraph discrepancy and the Erdős–Spencer/Erdős–Hajnal function F^(t)(n,α)
• Jumps and non-jumps of hypergraph Turán densities (Erdős's jumping constant conjecture, disproved by Frankl–Rödl)
• Hypergraph regularity / Gowers uniformity norms and density-increment arguments: quasirandom decomposition + counting/removal lemmas, and the iterative-density-increase route to Szemerédi-type theorems
• Reduced hypergraph method (Reiher–Rödl–Schacht) — using the hypergraph regularity/counting lemma to reduce uniform Turán density problems for k-graphs to finite extremal problems about reduced hypergraphs
• Incidence geometry: Szemerédi–Trotter theorem, the crossing lemma, and Zarankiewicz-type bounds
• P_t-free graphs: the Gyárfás path argument (χ-boundedness + minimal-counterexample cycle surgery)
• Kahn–Kalai expectation-threshold vs. threshold-probability conjecture (Park–Pham theorem)
• Kővári–Sós–Turán theorem: the double-counting bound ex(n,K_{s,t}) = O(n^{2-1/s})
• Landau–Ramanujan theorem: the density of sums of two squares is $\\Theta(x/\\sqrt{\\log x})$
• Lean 4 formalization of constructions and conditional reductions (Erdős-problem context)
• Log*-depth iterated/recursive lower-bound constructions (self-similar amplification budgeted by the inverse tower function)
• Logarithmic density: the $1/n$-weighted density that survives when natural density doesn't
• Lovász Local Lemma (symmetric, general/asymmetric, and algorithmic/random-recoloring variants) — probabilistic existence when bad events are individually non-negligible but sparsely dependent
• ℓp-norm (codegree-power-sum) reformulation of Turán density: maximize Σ(codegree)^p instead of edge count
• Structural constraints on a hypothetical minimal counterexample (minimum-degree, connectivity, adjacency pruning)
• nauty/geng/plantri: canonical-construction-path exhaustive generation of graphs (min-degree / cubic / planar families)
• Semi-random nibble vs. probabilistic log-prime/discrete-log constructions: two routes to beating the greedy exponent
• Nikiforov's counting-to-blowup theorem — a graph with a constant fraction of the possible copies of H contains a genuine blow-up H[t] with t = Θ(log n), and the open 'hypergraph Nikiforov' generalization
• Order type / orientation encoding of planar point sets (CC-systems, chirotopes, and the Aichholzer–Aurenhammer–Krasser database)
• Ordinal partition calculus — arrow notation $\\alpha\\to(\\alpha,m)^2$ and the self-partitioning-ordinal program
• Canonical-augmentation backtracking search for Erdős–Gyárfás minimal counterexamples (P13-free)
• Parity/pairing-halving argument for even-order $B_h$ liminf proofs
• Peres–Horodecki (PPT) criterion — partial transpose positivity is necessary for separability, and sufficient only in 2×2 and 2×3 usata 18×
• Perfect additive basis / unique representation basis ($r_A\\equiv1$)
• Singer perfect difference sets / finite-field constructions for Sidon-type lower bounds
• Additive-energy / pigeonhole averaging identity — $\sum_n r_A(n)=|A|^2$ forces large representation values on the dense side
• Pigeonhole reductions between Ramsey-type functions (e.g. H(k) ≥ W(k-1,k))
• Polynomial method / slice rank (Croot-Lev-Pach, Ellenberg-Gijswijt)
• Pór–Valtr positive-fraction Erdős–Szekeres theorem: dense clusters, not just points, in convex position
• PPT map — a channel that is CP and completely co-positive: Choi matrix PSD with positive partial transpose usata 6×
• Primitive sets (antichains in the divisibility poset: no element divides another)
• Machine formalization of infinitary combinatorics proofs (Isabelle/HOL, Lean)
• Ramsey-type concentration of the independence/clique number of G(n,1/2)
• Random algebraic construction (Bukh; Bukh–Conlon) — random low-degree polynomials over $\\mathbb F_q$ for Turán lower bounds
• Random quadratic form pseudorandom coloring constructions (Green–Hunter annulus method)
• Erdős–Simonovits rational exponents conjecture — every rational in [1,2] is a Turán exponent
• Reduce-and-measure before you spend — collapse the problem with the constraints in hand, size the reduced cost, verify the reduction is sound, THEN pick the method
• Reduction criterion — separable (and PPT) states obey ρ_A⊗I ⪰ ρ; the eigenvalue cap that killed our CCNR fitness usata 9×
• Large min/average degree forces a wide interval of cycle lengths (Liu–Montgomery robust-expander embedding)
• Additive representation function $r_{B,h}(n)$
• Rödl nibble / semi-random greedy method — iterated small-random-selection for near-perfect hypergraph matchings, packings, and colourings
• Ruzsa's log-of-primes probabilistic Sidon construction
• Ruzsa's number $R_m$ on $\\mathbb Z_m$ — the finite/periodic analogue of the Erdős–Turán additive-basis problem
• Ruzsa's prime-logarithm probabilistic Sidon-set construction and its discrete-log constructive analogue
• SAT/CP-SAT-based finite counterexample search and verification
• Second moment method / Paley–Zygmund inequality — variance-based random-structure existence proofs
• Semi-random method (Rödl nibble) — alias page; canonical content at concept/rodl-nibble
• Erdős–Hajnal shift graphs (explicit high-chromatic, controlled-odd-girth construction)
• Lindström-style shift/collision double-counting (upper-bound method for Sidon sets)
• Sidon sets / B_2 sets / Golomb rulers
• Sieve theory: Eratosthenes–Legendre, Brun, Selberg, Turán, and the large sieve
• Simulated-annealing extremal search — Metropolis local search for rare, structured witness/counterexample objects usata 12×
• Singer finite-field perfect difference set construction
• Baek–Balko split k-gon relaxation: an exactly-tight 2^{k-2}+1 threshold for a weakening of convex position
• R-spread set families (ALWZ/Rao's central reduction device)
• Stability method — bootstrapping an asymptotic extremal bound into an exact/unique result via 'near-extremal ⇒ structurally close to extremal'
• Supersaturation theorem — density strictly above the Turán threshold forces Ω(n^h) copies, not just one (Erdős–Simonovits 1983)
• Tame colouring profiles — smoothness/tail conditions making second-moment analysis of random-graph colourings tractable
• Topological Ramsey spaces / Nash-Williams tree-front machinery
• Turán number ex(n,H): extremal edge-count for forbidden subgraphs
• Uniform Turán density π_u(F) — Turán problem refined to uniformly-dense (quasirandom) host hypergraphs (Erdős–Sós, ~1990)
• Vertex-exposure martingale + Azuma–Hoeffding concentration for graph parameters
• Von Mangoldt-weighted Markov chains for primitive-set (Erdős) sums
• Erdős #1 — distinct subset sums need N ≫ 2^n
• Erdős #101 — four-point lines are $o(n^2)$ if no five on a line
• Erdős #1029 — polynomial-factor lower bound $R(k)/(k2^{k/2})\\to\\infty$
• Erdős #104 — o(n^2) unit circles through 3+ points
• Erdős #107 — exact Erdős–Szekeres convex-polygon constant
• Erdős #1083 — distinct distances in R^d (d≥3) is OPEN; the solved d=2 case (#89, Guth–Katz) is the literal numerical input to every known partial bound, via Solymosi–Vu's dimension-reduction recursion
• Erdős, Hajnal & Szemerédi (1982) — an $\\aleph_1$-chromatic graph with $h_G(n)=O(n^{3/2})$
• Erdős #113 — bipartite ex(n,G)≪n^{3/2} iff 2-degenerate: DISPROVED by Janzer 2021
• Erdős #1135 — the Collatz (3x+1) conjecture
• Erdős #114 — max lemniscate length is z^n−1
• Erdős #1145 — two-set a_n/b_n→1 additive-basis generalization is OPEN; its own necessity clause is settled by Erdős #331 (Ruzsa's digit-parity construction, DISPROVED)
• Erdős #1169 — ω₁²↛(ω₁²,3)² is solved under CH (Hajnal 1971); ZFC status open
• Does the K3 partition-ordinal property imply the Kn property? (Erdős–Hajnal, disproved by Darby/Schipperus/Larson)
• Erdős #119 — max modulus of unimodular-root products
• Erdős #1191 — how small can an infinite Sidon set's liminf density be?
• Erdős–Sárközy–Szemerédi (1966/1970) — primitive sets of large numbers: the tail Erdős-sum bound $\sum_{a\in A,\,a>x}1/(a\log a) < 1+o(1)$
• Erdős #120 — the Erdős similarity problem (affine copies)
• Erdős #1217 — Erdős–Sárközy–Szemerédi (1966) divisibility-chain density conjecture, solved 2026 via the von Mangoldt/zeta Markov-chain method
• Erdős #123 — three coprime bases give a d-complete set?
• Erdős #126 — must Σ prime factors of Π(a+b) beat log n?
• Erdős #128 — local edge density forces a triangle
• Erdős #132 — a second low-multiplicity distance
• Erdős #138 — does W(k)^{1/k} → ∞ for van der Waerden numbers?
• Erdős #143 — integer dilations $|kx-y|\\geq1$ force sparsity?
• Erdős #146 — $r$-degenerate bipartite $H$ forces $\\mathrm{ex}(n;H)\\ll n^{2-1/r}$
• Erdős #147 — min-degree-$r$ bipartite $H$ forces a Turán lower bound $n^{2-1/(r-1)+\\epsilon}$
• Erdős #158 (ancestor result) — the solved $g=1$ liminf-density theorem for infinite Sidon sets, and its bounded-multiplicity generalization
• Erdős #159 — polynomial improvement to R(C4,Kn) upper bound
• Erdős #161 — does hypergraph discrepancy-Ramsey jump only at 0?
• Erdős #164 — primes maximize the primitive-set sum Σ1/(n log n)
• Erdős #165 — asymptotic formula for the Ramsey number R(3,k)
• Erdős #18 — practical numbers: how few divisors suffice?
• Erdős #183 — growth rate of the k-colour triangle Ramsey number
• Erdős #190 — canonical Ramsey growth rate H(k)^{1/k}/k → ∞ (SOLVED)
• Erdős #2 — minimum modulus of a covering system cannot be arbitrarily large
• Erdős #20 — sunflower conjecture, exponential bound for f(n,k)
• Erdős #216 — the empty hexagon number g(6) (and non-existence for k≥7)
• Erdős #241 — sharp $N^{1/3}$ asymptotics for $B_3$ sets
• Erdős #28 — additive basis forces unbounded representations
• Erdős #30 — sharp $N^{1/2}$ asymptotics for Sidon sets
• Erdős #329 — how large can limsup |A∩[1,N]|/N^{1/2} be for a Sidon set?
• Erdős #39 — density $N^{1/2-\\epsilon}$ infinite Sidon set
• Erdős #4 — unbounded large gaps between primes (Rankin's constant removed)
• Erdős #40 — sharp density threshold for Erdős–Turán
• Erdős #41 — $B_3$ (distinct-triple-sum) set density liminf
• Erdős #42 — every Sidon set has a same-scale Sidon set with disjoint differences
• Erdős #43 — Sidon-pair size bound (disproved)
• Erdős #474 — 3-colouring $\\mathbb{R}^2$ vs. $2^{\\aleph_0}\\not\\to[\\aleph_1]_3^2$
• Erdős #50 — no point of positive derivative for the φ(n)/n distribution
• Erdős #500 — Turán density of the tetrahedron $K_4^{3}$
• Erdős #52 — the sum-product problem for integers
• Erdős #548 — Erdős–Sós conjecture: $\\frac{k-1}{2}n+1$ edges force every $(k+1)$-tree
• Erdős #552 — Ramsey number R(C4,Sn), dip below n+√n?
• Erdős #562 — Erdős–Hajnal–Rado hypergraph Ramsey growth rate: SOLVED at the ≥4-colour endpoint via the stepping-up lemma (2-colour case still open)
• Erdős #563 — the graph (t=2) discrepancy-Ramsey function F(n,α) has order Θ_α(log n)
• Erdős #564 — is $R_3(n) \\geq 2^{2^{cn}}$? (3-uniform hypergraph Ramsey)
• Erdős #588 — are $k$-rich lines $o(n^2)$ for $k\geq 4$?
• Erdős #590 — ω^ω → (ω^ω, 3)² (Chang's ordinal partition theorem)
• Erdős #591 — is $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$?
• Erdős #592 — characterize the countable partition ordinals
• Erdős #593 — characterize obligatory finite 3-uniform hypergraphs
• Erdős #595 — K4-free graph not a countable union of triangle-free graphs
• Erdős #601 — ordinal graphs: infinite path or full independent set
• Erdős #604 — pinned distinct-distances at a single point
• Erdős #625 — does χ(G)−ζ(G)→∞ for random G(n,1/2)?
• Erdős #63 — infinite chromatic number ⇒ cycles of length $2^n$ infinitely often
• Erdős #64 — min-degree-3 graphs contain a power-of-2 cycle usata 1×
• Erdős #65 (part 1) — Erdős–Hajnal: sum of reciprocal cycle lengths ≫ log(minimum degree)
• Erdős #651 — higher-dimensional Erdős–Szekeres convex-position numbers are subexponential for d≥3 (disproved)
• Erdős #66 — additive basis with $r_A(n)/\\log n \\to c\\neq 0$
• Erdős #661 — bipartite distinct-distances o(n/√log n) question is OPEN ($50); the underlying two-set function D(m,n) is SOLVED up to a log factor (Elekes construction 1995/1999, matching lower bounds by Mathialagan 2019)
• Erdős #671 — Lagrange interpolation despite blown-up Lebesgue function
• Erdős #687 — estimate the Jacobsthal covering function Y(x)
• Erdős #688 — the ε_n covering variant: how few large primes can cover [1,n]?
• Erdős #689 — double covering by prime residues: all-moduli case SOLVED (erdos/1205, F(x)~log x); primes-only case OPEN with two 2026 proof claims unverified
• Erdős #708 — linear bound for divisor-product covering constant g(n)
• Erdős #712 — Turán density of complete $r$-uniform hypergraphs $K_k^r$
• Erdős #713 — does every bipartite graph have a Turán exponent?
• Erdős #74 — almost-bipartite graph of infinite chromatic number
• Erdős #749 — upper-density variant solved (Bhalla 2026): bounded representation with sumset density → 1; lower-density case (adjacent to #28) still open
• Erdős #760 — large χ(G) forces a subgraph of large cochromatic number ζ(H) ≫ χ(G)/log χ(G)
• Erdős #762 — is χ(G)≤ζ(G)+2 when ω(G)<5 and ζ(G)≥4? (Erdős–Gimbel–Straight 1988)
• Erdős #77 — value of $\\lim_{k\\to\\infty}R(k)^{1/k}$
• Erdős #78 — constructive proof that $R(k)>C^k$
• Erdős #838 — f(n), the minimum number of distinct convex subsets of n points (open precise growth rate; two-sided quasi-polynomial bound SOLVED by Erdős 1978 via double counting against the Erdős–Szekeres theorem)
• Erdős #858 — max Erdős-sum over sets avoiding \"a·t=b, spf(t)>a\" chains
• Erdős #86 — C4-free subgraphs of the hypercube density
• Erdős #89 — distinct distances in the plane
• Erdős #90 — the unit distance conjecture (disproved 2026)
• Erdős #97 — convex polygon vertex with no 4 equidistant
• Jacobsthal's function h(k) — finiteness and near-matching two-sided bounds
• Erdős #99 — must a min-diameter unit-min-distance set contain a unit triangle?
• PPT² conjecture (Christandl 2012) — composition of two PPT channels is entanglement-breaking; open for dimension ≥ 4 usata 46×
• Ajtai–Komlós–Szemerédi (1981): first improvement on the trivial $N^{1/3}$ infinite Sidon bound, $A(x)\\gg(x\\log x)^{1/3}$
• AlphaEvolve-assisted optimization of the Carter–Hunter–O'Bryant Sidon upper-bound constant (Nov 2025)
• Borwein–Choi–Chu (2006) — the representation function of any order-2 additive basis cannot be bounded by 7
• Uniform Turán density of the broken tetrahedron K_4^{(3)-} equals 1/4 (Erdős–Sós problem, solved 2013–2016 by two independent methods)
• Cap-set problem — max progression-free subset of F_3^n
• Carr — any minimal Erdős–Gyárfás counterexample is predominantly cubic
• Chen (1993/1996) — $B_{2k}$-sequence liminf density collapse for every even $h=2k$
• Non-concentration of the chromatic number of G(n,1/2) (Heckel 2021)
• Cilleruelo (2012/2014) — an explicit, deterministic infinite Sidon set matching Ruzsa's exponent $x^{\\sqrt2-1+o(1)}$, via discrete logarithms
• Cilleruelo–Tesoro (2012/2015) — dense infinite $B_3,B_4$ sequences via Gaussian-prime arguments (construction side)
• Duffin–Schaeffer conjecture (1941) — the divergence criterion for metric Diophantine approximation by reduced fractions
• Erdős–Faber–Lovász conjecture, proved for all sufficiently large n (Kang–Kelly–Kühn–Methuku–Osthus, 2021/2023)
• Mubayi–Suk–Zhu 2020 — the penultimate case $t=k-1$ of the Erdős–Hajnal 1972 tower-growth conjecture, via an enhanced stepping-up lemma
• Conlon–Fox–Sudakov (2011/2012) — the Erdős–Hajnal conjecture has a tripartite 3-uniform analogue, and provably NO clique/independent-set analogue for k>3
• Erdős–Rényi (1960) — probabilistic construction of density-$N^{1/2-\\epsilon}$ sets with bounded representation function
• Erdős–Szekeres cup-cap lemma (1935) — the exact ordered-Ramsey number f(k,ℓ) = C(k+ℓ−4, k−2) + 1
• Erdős–Tetali theorem — existence of $\\log n$-representation ('economical') additive bases of every order $h$
• Konyagin–Lev: the Erdős–Turán problem, fully resolved for infinite abelian groups
• Erdős #64 — computational bound extended: no counterexample on ≤ 19 vertices (2026-07-02)
• Erdős #64 — full exclusion campaign 2026-07-02: n≤19, GP≤2000, no necklace gadget ≤18, high-girth named graphs
• Erdős #64 — unconditional 4-or-8 dichotomy ≤19, first C4+C8-free specimens (n=24), Pt-line ceiling at P17
• Erdős #64 — no vertex-transitive counterexample up to 1280 vertices (census sweep, 2026-07-02)
• Finite-field Kakeya conjecture (Wolff 1999, proved by Dvir 2008) — the polynomial method's founding application
• Alexeev–Mixon (2025) — a finite Sidon set that cannot be extended to a finite perfect difference set (Erdős #707 disproved)
• Frieze's trick — boosting a positive-probability (second-moment) existence bound to whp via the vertex-exposure martingale
• Erdős–Gyárfás conjecture confirmed for 3-connected cubic planar graphs (Heckman–Krakovski 2013)
• Conlon–Fox–Sudakov 2010 — off-diagonal 3-uniform hypergraph Ramsey numbers $r_3(s,n)$: a near-tight upper bound via an online-Ramsey-game reduction, and the first superexponential lower bound
• Independence/clique number of G(n,1/2) is two-point concentrated at ⌊α₀+o(1)⌋
• Kahn–Kalai conjecture — threshold vs. expectation-threshold for monotone properties
• Konstantoulas (2013) — near-cofinite sumset forces representation function $>5$ infinitely often
• Alon–Krivelevich–Sudakov (2003) — Turán number of bipartite $H$ with bounded degree on one side, $\\mathrm{ex}(n,H) = O(n^{2-1/r})$
• Large almost monochromatic subsets in hypergraphs (Conlon–Fox–Sudakov, 2009) — resolves the t=3 case of Erdős–Hajnal's 1989 discrepancy problem (Erdős #161)
• Liu–Montgomery (2020/2023) — Erdős–Hajnal odd-cycle reciprocal-sum problem
• Markström–Royle exhaustive computer search: lower bounds on Erdős–Gyárfás counterexamples
• Nash (1989) — $B_4$-sequence liminf density collapse: $\\liminf_n A(n)\\log^{1/4}n/n^{1/4}<\\infty$
• Du–Hu–Liu–Wang (Apr 2026), improved by Fan–Li–Lin–Ning (May 2026) — a double-exponential lower bound for $r_4(5,n)$, completing the Erdős–Hajnal 1972 off-diagonal hypergraph-Ramsey tower-growth-rate program
• Off-diagonal van der Waerden numbers — W(3,k) grows superpolynomially, disproving Graham's O(k²) conjecture
• Erdős–Gyárfás conjecture confirmed for P8-free and P10-free graphs
• Pliego 2024 — sharp density-vs-boundedness trade-off for $B_2[g]$ sequences
• PPT² iteration 1 — WH-mixture exclusion at n=4; first bound-entangled CHANNEL built and composed (n=3 landscape) usata 6×
• PPT² iteration 10 — the DSE theorem (first proven partial result); the key∘key hard core is DPS-2-extremal; the DPS-3 razor usata 16×
• PPT² iteration 11 — razor verdict: DPS-3 feasible at slack exactly 0; and the hard core is secretly a PAULI CHANNEL (Bell-diagonal, Z[√2]) usata 8×
• PPT² iteration 12 — THEOREM: the entire HPHH pbit family satisfies PPT² exactly (Ψ_θ∘Ψ_θ is EB, explicit algebraic certificates); the 'extremality' of iters 10–11 honestly deflated usata 13×
• PPT² iteration 13 — coverage explodes: 87/87 mixed pbit pairs EB; twisted-square lemma + 24 new channels (18 new BE); the residual set is named usata 7×
• PPT² iteration 14 — the twisted kernel polytope closes the residual: the ENTIRE named zoo is certified EB; the criterion is COMPLETE on its algebra usata 3×
• PPT² iteration 15 — exhaustiveness mapped: PPT²(n=4) ⟺ EB at extreme×extreme; 6-orbit skeleton with universal complete LP; 258,521 certificates; the one missing invention named usata 1×
• PPT² iteration 2 — first n=4 ghost channel; two no-gos (padding, adjoint diagonal); the hunt becomes transfer-matrix alignment usata 3×
• PPT² iterations 3–4 — composed CCNR pins at 1; resolved: realignment is PROVABLY blind to PPT² (shadow theorem, 3-line proof) usata 8×
• PPT² iteration 5 — DPS-2 silent at the CCNR frontier; majorization criterion also provably blind usata 3×
• PPT² iteration 6 — DPS-2 blindness ⟺ antidegradability: key-repeater bridge found; all four structured constructions honestly refuted usata 9×
• PPT² iteration 7 — reading the SDP's extension: fat interior, mid-rank extremals, coherence-distance hypothesis REFUTED usata 3×
• PPT² iteration 8 — pbit-seeded hunt: one composition destroys the key structure; even the necessary condition fails on the best possible seeds usata 12×
• PPT² iteration 9 — adversarial gap maximization pins at the solver floor: the counterexample front is exhausted; campaign commits to the proof usata 4×
• Erdős–Simonovits rational exponents conjecture — the single-graph case near exponent 2 (Conlon–Janzer 2022)
• Rödl 1982 — the hypergraph analogue of Erdős–Hajnal–Szemerédi's almost-bipartite large-chromatic-number problem, SOLVED (the graph case, erdos/74, remains open)
• Ruzsa (1990) — a basis of order 2 with representation function bounded in square mean, $\\sum_{n\\le N}\\sigma_A(n)^2=O(N)$
• Ruzsa's probabilistic infinite Sidon set, $A(x)=x^{\\sqrt2-1+o(1)}$ (1998)
• Ruzsa's infinite Sidon set of density $x^{\\sqrt2-1+o(1)}$ (1998) — current record
• Ruzsa's number $R_m$: every finite cyclic group has an absolutely-bounded-representation basis (Ruzsa, 1990)
• Singer's finite-field perfect difference set construction (1938)
• Talagrand's fractional expectation-threshold conjecture (proved by Frankston–Kahn–Narayanan–Park, 2019)
• Turán's tetrahedron conjecture solved in the ℓ2-norm: codegree-squared-sum density σ(K₄³) = 1/3, uniquely extremal (Balogh–Clemen–Lidický 2021/2022; uniqueness by Bodnár–Chen–Deng 2025)
• Fox–Hunter (2026) — three-color van der Waerden numbers $w(k;3)$ grow super-exponentially in $k$
• Turán and Sidorenko bounds for tree-degenerate bipartite graphs (Grzesik–Janzer–Nagy 2019; Jiang–Longbrake 2022)
• Grzesik–Janzer–Nagy (2019/2022) — the Turán number of blow-ups of trees: $\\mathrm{ex}(n,T[r])=O(n^{2-1/r})$, the $r$-degenerate blow-up-of-a-tree case of Erdős's 1967 degenerate-Turán conjecture
• DeepMind/Wagner/Bloom (2026) — the van der Waerden difference sub-question: $W(k+1)\\geq W(k)+k$
• Gowers (2001) — explicit tower-type upper bound on van der Waerden numbers W(k), via a new Fourier-analytic proof of Szemerédi's theorem
• Kozik–Shabanov (2016) — algorithmic Lovász Local Lemma (entropy compression) gives the van der Waerden lower bound W(k,r) > c·r^{k-1}