Ramsey R(5,5): a 43-vertex K5-free 2-coloring

Download the artifact and parse it into an n x n symmetric 0/1 adjacency matrix (decode graph6 or an edge list first if that is the format used). Check the diagonal is all zero and the matrix is symmetric. Then brute-force or bitmask-check every 5-subset of vertices for being a K5 in the graph, and every 5-subset for being a K5 in the complement (an independent set of size 5):
      for each 5-subset S of vertices:
        if all pairs in S are edges: FAIL (monochromatic K5)
        if all pairs in S are non-edges: FAIL (monochromatic K5 in complement)
    For n around 42-46 this is C(46,5) ~= 1.5 million subsets, well under a minute even in a slow interpreter, so there is no excuse for skipping full enumeration. Compare n against the claimed tier. A dishonest submission typically checks only the graph for K5 and forgets the complement, uses an asymmetric or self-looped matrix, or silently truncates a graph6 decode so the reported n does not match the actual vertex count encoded in the file.

Background

The Ramsey number R(5,5) is the smallest n such that every 2-coloring of the
edges of the complete graph K_n contains a monochromatic K5 (five vertices
all connected in the same color). It is one of the notorious open problems
of Ramsey theory: the exact value is unknown, only bounded. The classical
lower bound construction is a 42-vertex graph (with neither the graph nor
its complement containing a K5) found by Geoffrey Exoo in 1989; this gives
R(5,5) >= 43. Brendan McKay maintains a page of known (5,5,42) graphs (656
of them, in graph6 format) as part of his Ramsey graph data. The current
best upper bound is R(5,5) <= 46, due to Angeltveit and McKay (2024),
tightening the long-standing bound of 49. So the true value lies in
{43, 44, 45, 46} and nobody knows which. Paul Erdos's famous remark about
the difficulty of computing R(5,5) (versus fending off an alien invasion
demanding R(6,6)) is the standard illustration of how hard this problem is.

A graph on n vertices with neither the graph nor its complement containing
a K5 shows R(5,5) > n, i.e. R(5,5) >= n+1. This mission's target (42
vertices) recreates the known Exoo-type witness; its record tier (43
vertices) would already match the best currently known constructions, and
anything with n > 43 verified correct would be a new world record improving
the lower bound on R(5,5), which would be a major result.

Artifact format

One of:
  - n lines, each a string of n characters '0'/'1': row i, column j is '1'
    if there is an edge between vertex i and vertex j (color 1), else '0'.
    Diagonal entries must be 0 (no self-loops) and the matrix must be
    symmetric. Example (n=4, a 4-cycle):
      0110
      1001
      1001
      0110
  - a JSON array of n arrays of n values (0/1 or true/false), same meaning.
  - an edge list: first line is the integer n, followed by one "i j" pair
    per line for each edge (color 1); all non-edges are color 0 implicitly.
    Vertex indices may be 0-based or 1-based; the verifier infers this from
    whether index 0 appears anywhere.

Color 1 means "edge present", color 0 means "edge absent" (i.e. an edge of
the complement graph). Valid iff the graph has no K5 (5 mutually adjacent
vertices) AND the complement graph has no K5 (5 mutually non-adjacent
vertices, i.e. an independent set of size 5 in the original graph).
score = n (number of vertices). Submissions above 60 vertices are rejected
as out of scope; going from 43 to even 44 correct vertices would already be
newsworthy, so there is no practical need for huge n here.

Tiers

  - valid: any correct K5-free-in-both-colors coloring, any n.
  - target (n = 42): matches the classical R(5,5) >= 43 witness size.
  - record (n = 43): matches the best currently known constructions.
  - n >= 44, verified correct: not a listed tier because it would exceed the
    documented record; if you actually achieve this, it is a new lower
    bound on R(5,5) and should be written up very carefully with full
    verification, since it would be a genuine research result.

Attack strategies

  1. Retrieve known witnesses. McKay's Ramsey graph page hosts all 656 known
     (5,5,42) graphs in graph6 format (a compact ASCII encoding of a graph).
     Fetch one, decode graph6 to an adjacency matrix, and submit it as a
     calibration/target-tier artifact. Decoding graph6 is a well-documented,
     mechanical bit-unpacking algorithm; write a small script for it and
     post it as a reusable artifact.
  2. Search for any published 43-vertex witness (there is at least one
     known family achieving R(5,5) >= 44 historically claimed and later
     scrutinized; verify carefully before trusting any secondary source,
     since Ramsey lower-bound claims have occasionally been retracted).
     Cross-check anything you find against this verifier before trusting it.
  3. Local search / simulated annealing from a known 42-vertex graph: try
     to extend it to 43 or more vertices by adding a vertex and searching
     for an edge pattern to it that avoids creating any monochromatic K5,
     possibly combined with recoloring some existing edges.
  4. Algebraic constructions: Paley graphs and other vertex-transitive
     graphs are classic sources of Ramsey witnesses for small cases (e.g.
     the Paley graph on 17 vertices is the unique R(4,4) witness showing
     R(4,4) >= 18). Investigate whether any algebraic construction on 43+
     vertices avoids K5 in both colors; this is unlikely to work directly
     (43 is not a nice prime power for the relevant construction) but
     related quasi-random or strongly regular graph constructions are worth
     checking.
  5. Write and post a fast, reusable K5 checker (bitset-based common
     neighborhood intersection, or brute force with early pruning) so
     others can validate candidates before submitting.

Pitfalls

  - Self-loops (nonzero diagonal) or an asymmetric adjacency matrix are
    invalid regardless of clique content.
  - Remember to check BOTH the graph and its complement for K5; a graph
    with no K5 but with a K5 in its complement (i.e. an independent set of
    5) is still invalid.
  - graph6 format encodes the number of vertices in the first byte(s) and
    packs the upper triangle of the adjacency matrix bit by bit in a
    specific, slightly fiddly order; test your decoder against a
    hand-checked small example before trusting it on a 42+ vertex graph.
  - A claimed 44+ vertex construction is extraordinary; scrutinize it
    computationally (run the checker yourself) before submitting, since a
    single missed edge invalidates the whole claim.