A triangular book graph 
consists of 
triangles sharing
one common graph edge. Equivalently, it is the graph
join 
of a complete graph on two vertices
and an empty graph on 
vertices. It has 
vertices and 
edges. This is the book graph
of the Ramsey number literature, distinct from the
book graph 
formed from squares (Rousseau and Sheehan
1978).
For these triangular books, 
is the least 
such that every red-blue edge
coloring of 
contains a red 
or a blue 
.
Chung and Wesley (2026) proved
 |
whenever 
or 
is the order of a conference graph. In particular,
either number may be a prime power congruent to 1
modulo 4. They also proved the equality when 
for a prime power 
with 
, and when 
for a prime power 
with 
. Chung and Wesley (2026) report that an AI system
assisted in discovering several of these constructions.
