Where did Erdős propose that a simple nonbipartite graph with $m$ edges and $n$ vertices contains a triangle when $m>\frac{1}{4}(n-1)^2+1$?

Question

The following theorem seems to be well-known; it can be found in Bondy and Murty's book Graph Theory (GTM 244,Page 306, Exercise 12.2.7), where the authors of the book attribute it to Erdős, but do not specify the source. I would like to inquire about which article by Erdős obtains it.

Theorem 1. Let $G$ be a simple nonbipartite graph with $m>\frac{1}{4}(n-1)^2+1$ (where $m$ and $n$ are the numbers of edges and vertices, respectively). Then $G$ contains a triangle.

P.S. we can see the proof in Prove that : $f(n) \le \frac{1}{4}(n-1)^2+1 $

$m$ and $n$ are the number of edges and vertices of $G$, respectively — licheng, Jan 17 '24 at 13:06
In this answer, I talk about some articles with related results. Erdős (along with Faudree and Pach) does prove a theorem that might imply this (I haven't checked), but in any case, Brauwer already proved this precise theorem in 1981, so I think the attribution to Erdős is mistaken. Does that answer your question? — Misha Lavrov, Jan 17 '24 at 15:26
@MishaLavrov Thank you. Through my research, this result should be attributed to Erdős and has been rediscovered multiple times. I will write an answer. — licheng, Jan 18 '24 at 02:56

licheng · Accepted Answer · 2024-01-18T08:20:36.790

Through my research, this result came from the paper below.

Erdős, On a theorem of Rademacher-Turán. Illinois J. Math, 1962, 122-127.

We can download in https://static.renyi.hu/~p_erdos/1962-09.pdf.

Lemma 1 in the paper is the result we are referring to. In addition, Erdős said that the result was found jointly by Gallai and himself and also found by Andrásfai independently. So in a paper titled Strong Turán Stability, the authors stated that the result was proven by Andrásfai, Erdős, and Gallai.

It has been independently re-discovered several times. For example,

Amin, K., J. Faudree, R. J. Gould and E. Sidorowicz, On the non-(p −1)-partite Kp-free graphs, Discuss. Math. Graph Theory 33 (2013), pp. 9–23.
Hanson, D. and B. Toft, k-saturated graphs of chromatic number at least k, Ars Combin. 31 (1991), pp. 159–164.
Kang, M. and O. Pikhurko, Maximum Kr+1-free graphs which are not r-partite, Mat. Stud. 24 (2005), pp. 12–20.

I feel like I've been accused of something, but I'm not sure what. — Misha Lavrov, Jan 18 '24 at 07:34
@MishaLavrov No, I have no charges, haha. Thank you! You have helped me a lot, always. I asked this because I was unsure (as someone just cites other papers) — licheng, Jan 18 '24 at 08:15

Where did Erdős propose that a simple nonbipartite graph with $m$ edges and $n$ vertices contains a triangle when $m>\frac{1}{4}(n-1)^2+1$?

1 Answers1