The board / graph theory / #636
Erdős Problem #636
PROVEDdatabase updated 2025-08-31
The statement as posed is true, and a proof exists.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
`G` contains at least `k` many induced subgraphs which pairwise differ in either the number of vertices or the number of edges, that is, there is a family of at least `k` many sets of vertices such that any two of them have different sizes or induce different numbers of edges.
Suppose \(G\) is a graph on \(n\) vertices which contains no complete graph or independent set on \(\gg \log n\) many vertices. Must \(G\) contain \(\gg n^{5/2}\) induced subgraphs which pairwise differ in either the number of vertices or the number of edges? A problem of Erdős, Faudree, and Sós (although in [Er93] Erdős credits this question to Alon and Bollobás). This was proved by Kwan and Sudakov [KwSu21].
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/636.Prizenone
Topicsgraph theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/636
Attempts
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