OpenFrontiers
Join
← Back to the index
OP-9300OpenGraph theory

Erdős #1011 · graph theory

Let fr(n)f_r(n) be minimal such that every graph on nn vertices with ≥fr(n)\geq f_r(n) edges and chromatic number ≥r\geq r contains a triangle. Determine fr(n)f_r(n).

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let fr(n)f_r(n) be minimal such that every graph on nn vertices with ≥fr(n)\geq f_r(n) edges and chromatic number ≥r\geq r contains a triangle. Determine fr(n)f_r(n).

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Turán's theorem implies f2(n)=⌊n2/4⌋+1f_2(n)=\lfloor n^2/4\rfloor+1. Erdős and Gallai [Er62d] proved f3(n)=⌊14(n−1)2⌋+2f_3(n)=\lfloor \frac{1}{4}(n-1)^2\rfloor+2.

Simonovits showed in his PhD thesis (see the discussion on p. 358 of [Si74]) that

fr(n)=n24−g(r)2n+O(1),f_r(n)=\frac{n^2}{4}-\frac{g(r)}{2}{n}+O(1),

where g(r)g(r) is the largest mm such that, for any triangle-free graph with chromatic number ≥r\geq r, at least mm vertices of GG need to be removed to obtain a bipartite graph. Simonovits [Si74] notes

log⁡rlog⁡log⁡rr2≪g(r)≪(log⁡r)2r2.\frac{\log r}{\log\log r}r^2 \ll g(r) \ll (\log r)^2r^2.

Hunter in the comments has noted that other results imply g(r)≍r2log⁡rg(r)\asymp r^2\log r - in fact

(1/2−o(1))r2log⁡r≤g(r)≤(2+o(1))r2log⁡r.(1/2-o(1))r^2\log r\leq g(r)\leq (2+o(1))r^2\log r.

The lower bound follows from work of Davies and Illingworth [DaIl22] (see [1104]). The upper bound follows from work of Hefty, Horn, King, and Pfender [HHKP25] on R(3,k)R(3,k).

Ren, Wang, Wang, and Yang [RWWY24] showed that, for n≥150n\geq 150,

f4(n)=⌊(n−3)24⌋+6.f_4(n)=\left\lfloor\frac{(n-3)^2}{4}\right\rfloor+6.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1011 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.