The board / graph theory / #127
Erdős Problem #127
PROVED LEANdatabase updated 2026-08-24
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`f m` is the largest `k` such that every graph with `m` edges contains a bipartite subgraph with at least \(\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+k\) edges.
Let \(f(m)\) be maximal such that every graph with \(m\) edges must contain a bipartite graph with \[\geq \frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+f(m)\] edges. Is there an infinite sequence of \(m_i\) such that \(f(m_i)\to \infty\)? Conjectured by Erdős, Kohayakava, and Gyárfás [Er97b]. Edwards [Ed73] proved that \(f(m)\geq 0\) always. Note that \(f(\binom{n}{2})= 0\), taking \(K_n\). Solved by Alon [Al96], who showed \(f(n^2/2)\gg n^{1/2}\), and also showed that \(f(m)\ll m^{1/4}\) for all \(m\). The best possible constant in \(f(m)\leq Cm^{1/4}\) is unknown.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/127.Prizenone
Topicsgraph theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/127
Attempts
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