The Burr-Erdős conjecture asserts that, for each fixed nonnegative integer 
,
there is a constant 
such that every simple graph 
on 
vertices and with graph
degeneracy at most 
has Ramsey number at most
(Burr and Erdős 1975). Here the Ramsey number
is the least 
such that every two-coloring of the edges of 
contains a monochromatic copy of 
. A graph of graph
degeneracy at most 
has a graph vertex of vertex
degree at most 
in every nonempty subgraph.
Lee (2017) proved the conjecture. Thus bounded degeneracy suffices for a Ramsey number that is linear in the number of vertices, even when the maximum vertex degree is unbounded.
This is problem 163 in the Erdős problems collection (Bloom 2026).
