An exact-distance Hamming graph 
is the graph on words
of length 
over an alphabet of 
symbols in which two words are adjacent precisely when their Hamming
distance is 
, where 
, 
, and 
are integers with 
, 
, and 
. It is the graph
distance graph at distance 
of the Hamming graph 
.
In particular, 
, whereas larger exact distances generally give
different graphs (Brešar et al. 2019). Each
graph vertex has vertex
degree 
,
by choosing the positions that change and one of 
replacement symbols at each position.
Let 
be the largest number of vertices whose induced
subgraph is a triangle-free graph. Csajbók
et al. (2026) proved that 
and 
for every fixed integer 
.
They also determined the exact order in the ternary distance-four case,
 |
where 
is big-Theta notation and the constants
implicit in 
may depend on 
. The asymptotic estimates are for 
.
Csajbók et al. (2026) report that ChatGPT 5.6 Sol assisted them in obtaining the main results by building on their mathematical ideas.
