The first record of `r55_42some.g6` decodes to a graph \(G\) on 42 vertices with 425 edges. Exhaustive enumeration of its \[ \binom{42}{5}=850{,}668 \] five-vertex subsets gives the induced-edge histogram \[ [0,6078,31577,99576,192531,225712,173847,87463,27923,5961,0]. \] The entries at positions 0 and 10 vanish. Thus every five-set has at least one edge and at least one nonedge. The graph contains no \(K_5\), and its complement contains no \(K_5\).

By definition, \(R(5,5)\) is the least order at which every graph contains a \(K_5\) or an independent five-set. This 42-vertex witness proves \[ R(5,5)>42,\qquad R(5,5)\geq43. \] A 43-vertex witness would instead prove \(R(5,5)\geq44\). The candidate asks for that larger witness. Its cited graph and digest have order 42.

The candidate contains no graph6 record or adjacency matrix of order 43. Its computation identifies the first record in `r55_42some.g6`. The leading graph6 byte confirms order 42.

McKay and Radziszowski reported 656 known \((5,5,42)\)-graphs in section 4 of their 1997 paper. They checked that those graphs could not be extended by one vertex and conjectured both \(R(5,5)=43\) and completeness of the 656-graph collection. The completeness statement remains a conjecture in that paper, so the failed extensions do not exclude every possible 43-vertex graph.

Angeltveit and McKay's 2026 paper states that Exoo's lower bound 43 remains the best and proves the current upper bound 46. McKay's data page likewise says that graphs with 43 through 47 vertices could exist. The current interval is \[ 43\leq R(5,5)\leq46. \] An explicit \((5,5,43)\)-graph would improve its lower endpoint to 44. No such artifact was supplied or located in the checked primary sources.

The graph6 short header is the byte `i`. Since \(\operatorname{ord}(i)-63=42\), the record has 42 vertices. Its 144 payload characters supply 864 bits. The first \(\binom{42}{2}=861\) bits encode the unordered pairs in graph6 order, and the final three bits are zero padding.

The graph6-line SHA-256 includes the terminating line feed. For a second digest, the decoder visits edges in graph6 order and serializes each edge \((u,v)\) as the two bytes `u,v`. The 425 edges therefore give 850 digest bytes.

The executable counts the number of induced edges on each five-set. Ten edges would be a \(K_5\) in the graph, while zero edges would be a \(K_5\) in the complement. Both counts are zero. The complete histogram also fixes every intermediate count and makes the check reproducible without a graph library.