AI Proof WatchKeep up with AI contributions

Lower bounds for finite two-colour Ramsey numbers R(s,t)

The two-color Ramsey number R(s,t) is the least vertex count forcing an s-clique or an independent t-set. Lower bounds require graphs avoiding both; explicit constructions seek formulas as well as computer-found finite witnesses.

Why it matters and what counts as progress

Why it matters: New witnesses improve concrete Ramsey bounds, while algebraic constructions can provide families with controlled clique and independence numbers.

What counts as progress: A verified graph raising a listed R(s,t) lower bound, or a direct explicit family improving the relevant asymptotic comparison under the same construction model.

How the problem developed

  1. 21 Apr 2026 · Computational lower bounds [1]

    Ansh Nagda, Prabhakar Raghavan and Abhradeep Thakurta Reported improved lower bounds for nine finite two-colour Ramsey numbers from computer-generated constructions.

  2. 22 Aug 2026 · Asymptotic lower bound [2]

    Ferdinand Ihringer and Sam Mattheus Proved a uniform lower bound R(s,t)≥t^((1−o(1)) log s/log(log s+1)) for 3≤s≤t, including a stated improvement for R(33,t).

AI contributions 2

An improved algebraic construction for Ramsey numbers

Ferdinand Ihringer, Sam Mattheus

ChatGPT 5.6 suggested the two-way trace graph TG_{d,h} of Remark 2.2 on projective points over F_(2^h). Ihringer and Mattheus bound its clique size by binom(d−2,ceil((d−2)/2)) and independence number by (d^h+1)². Their stronger one-way ordered graph and Theorem 1.3 are human-developed refinements.

AI involvementAI-centered
ImportanceNotable
Technical verificationFormal (human)
Human understandingFull
Paper date:

Read the contribution and its assessment →

Sources

  1. Ansh Nagda, Prabhakar Raghavan and Abhradeep Thakurta, Reinforced Generation of Combinatorial Structures: Ramsey Numbers
  2. Ferdinand Ihringer and Sam Mattheus, An improved algebraic construction for Ramsey numbers
  3. Ramsey graph paper and AI declaration
  4. Ramsey construction listing
  5. Nagda, Raghavan and Thakurta’s Ramsey paper
  6. Released Ramsey matrices and verifier
Website timeline

Publication dates and editorial changes.