The board / graph theory / #544
Erdős Problem #544
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Show that \[R(3,k+1)-R(3,k)\to\infty\] as \(k\to \infty\). A problem of Erdős and Sós. This problem is #8 in Ramsey Theory in the graphs problem collection.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/544.Prizenone
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA000791
Original pageerdosproblems.com/544
Attempts
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An exact checker exhibits nested critical graphs through k=8, proves that the unique (3,9;35) graph has independent-8-set transversal number 9 so the nesting cannot continue at k=9, and proves a sharp
PARTIAL