{
  "problem": "Erdos Problem 835 (Erdos-Rosenfeld)",
  "statement": "Does there exist k>2 such that the k-subsets of [2k] can be (k+1)-coloured so that every (k+1)-subset sees all k+1 colours among its k-subsets?",
  "equivalent_form": "chi(J(2k,k)) = k+1 for the middle Johnson graph.",
  "date": "2026-07-23",
  "authors": ["Nathan Wilbanks", "Annie"],
  "affiliation": "AGNT Labs",
  "status_of_problem": "OPEN overall; smallest unresolved case k=16.",
  "claims": [
    {
      "id": "C1",
      "claim": "835(k) YES iff chi(J(2k,k)) = k+1 (every anti-star (k+1)-clique rainbow <=> proper (k+1)-colouring).",
      "type": "reduction (known; matches formal-conjectures property_iff_chromaticNumber)",
      "evidence": "settle_835.py encodes exactly this and re-verifies rainbow-on-all-cliques for the k=2 witness.",
      "attribution": "erdosproblems.com/835 (Bloom); formal-conjectures 835.lean"
    },
    {
      "id": "C2",
      "claim": "Each colour class is a distance-4 constant-weight code with <= C_k = Catalan(k) codewords, equality iff it is a Steiner system S(k-1,k,2k). Since (k+1)C_k = binom(2k,k), 835(k) YES iff the middle layer partitions into k+1 Steiner systems S(k-1,k,2k) (large set).",
      "type": "theorem (Theorem 1); exact packing argument",
      "evidence": "Catalan divisibility and (k+1)C_k = binom(2k,k) checked exactly in sieve_835.py.",
      "attribution": "code/Johnson-bound framing known (forum: eigensolver, Tang). Explicit large-set characterization: this note."
    },
    {
      "id": "C3",
      "claim": "The Steiner divisibility conditions for S(k-1,k,2k) hold iff k+1 is prime. Verified for all 2<=k<=300 with zero mismatches. Hence composite k+1 => no Steiner system => A(2k,4,k) <= C_k - 1 => chi >= k+2.",
      "type": "theorem (Theorem 2); machine-verified over a range",
      "evidence": "sieve_835.py: survivors == {k : k+1 prime}; mismatches == NONE on 2..300. Odd-k obstruction lambda_{k-2}=(k+2)/2 proven in closed form.",
      "attribution": "Independent elementary route to the Ma-Tang composite direction [MT25]."
    },
    {
      "id": "C4",
      "claim": "When the Steiner system is absent, ceil(binom(2k,k)/(C_k - 1)) = k+2 exactly: one missing codeword forces chi >= k+2.",
      "type": "exact arithmetic fact",
      "evidence": "sieve_835.py strict-gap check for k in {3,5,7,8,9,10,16,112}: all give k+2.",
      "attribution": "this note"
    },
    {
      "id": "C5",
      "claim": "Recursive Johnson bound jb(2k,4,k) = C_k iff k+1 is prime (3<=k<=60); strictly smaller (hence 835 NO) otherwise. Independent corroboration of the {k+1 prime} hard core.",
      "type": "computation",
      "evidence": "johnson_bound_835.py: undecided-by-bound set == {k : k+1 prime}.",
      "attribution": "recursion as in formal-conjectures 835.lean; corroboration by this note"
    },
    {
      "id": "C6",
      "claim": "835(2) is YES.",
      "type": "settled (witness)",
      "evidence": "settle_835.py: own prover finds a colouring (4 nodes); verified proper AND every (k+1)-set rainbow; CaDiCaL/Glucose/MiniSat all SAT. witness_k2.json.",
      "attribution": "known trivial case; independent verification here"
    },
    {
      "id": "C7",
      "claim": "835(3) is NO (J(6,3) not 4-colourable; STS(6) nonexistence).",
      "type": "settled (five-oracle + design nonexistence)",
      "evidence": "settle_835.py (9 nodes) and crosscheck_nosym.py (257 nodes, no symmetry break); 3 SAT solvers UNSAT; lambda_1=5/2.",
      "attribution": "known (Brouwer); independent certificate here"
    },
    {
      "id": "C8",
      "claim": "835(4) is NO (J(8,4) not 5-colourable). Smallest structural case: SQS(8) exists but no large set of five disjoint SQS(8) tiles the 70 quadruples.",
      "type": "settled (five independent oracles)",
      "evidence": "settle_835.py own prover 136 nodes with symmetry break; crosscheck_nosym.py DSATUR 11486 nodes with NO symmetry break, 0.46s; CaDiCaL, Glucose4, MiniSat22 all UNSAT on J_8_4__5col.cnf (350 vars, 3575 clauses).",
      "attribution": "known (Brouwer); this note gives a self-contained five-oracle certificate"
    },
    {
      "id": "C9",
      "claim": "835(5) is NO (S(4,5,10) fails divisibility, lambda_2 = 28/3).",
      "type": "settled (divisibility)",
      "evidence": "sieve_835.py.",
      "attribution": "this note (consistent with known NO for 3<=k<=8)"
    },
    {
      "id": "C9b",
      "claim": "835(6) is NO (J(12,6) not 7-colourable). The case Erdos and Rosenfeld were unsure about; S(5,6,12) exists but admits no large set.",
      "type": "settled (direct SAT)",
      "evidence": "settle_k6b.py: CaDiCaL UNSAT on the 924-vertex, 7-colour instance in 328s with one anti-star clique fixed to identity. settle_k6_result.json. Independent of Brouwer's tabulated value.",
      "attribution": "known NO (Brouwer); this note gives an independent SAT certificate"
    },
    {
      "id": "C10",
      "claim": "Open frontier is exactly {k = p-1 : p prime} with S(k-1,k,2k) existence and large-set tiling unresolved; smallest open case k=16, equivalent to deciding A(32,4,16) vs C_16 / existence of S(15,16,32).",
      "type": "frontier statement (honest)",
      "evidence": "Theorem 2 survivors + Ma-Tang; Tang brute force leaves {16,112,256,268}.",
      "attribution": "Ma-Tang [MT25]; Tang forum note; framing by this note"
    }
  ],
  "not_claimed": [
    "We do NOT claim to resolve 835 (the existence question remains open).",
    "We do NOT claim the reduction to the Johnson graph or the composite-case theorem; those are prior work (Bloom; Ma-Tang).",
    "We do NOT claim k=16 or any k+1-prime case beyond the settled small ones."
  ]
}