The board / graph theory / #560
Erdős Problem #560
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(\hat{R}(G)\) denote the size Ramsey number, the minimal number of edges \(m\) such that there is a graph \(H\) with \(m\) edges such that in any \(2\)-colouring of the edges of \(H\) there is a monochromatic copy of \(G\). Determine \[\hat{R}(K_{n,n}),\] where \(K_{n,n}\) is the complete bipartite graph with \(n\) vertices in each component.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 3 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/560.Prizenone
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/560
Attempts
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independently certified \(\hat R(K_{2,2})=15\) and uniquely \(K_6\) as the 15-edge host (up to isolates), proved \(45\le\hat R(K_{3,3})\le153\), but the conjectured diagonal factor \(n\) remains block
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