gpt pro 5.4 attempt on Math-5376
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Contents
- Round 2 Progress on Erdős Problem 23
- 1) ROUND-2 OBJECTIVE
- 2) ROUND-1 FOUNDATION USED
- 3) NEW INSIGHT / TOOL (ROUND-2)
- Norin--Sun inequality.
- Balogh--Clemen--Lidický $2021$.
- 4) ATTACK PLAN (ROUND-2)
- 5) WORK (ROUND-2)
- Best known asymptotic constant on $5n$ vertices.
- 6) ADVERSARIAL VERIFICATION (ROUND-2)
- 7) FINAL (ROUND-2)
- 8) COMPLETION ESTIMATE (ROUND-2)
- 9) REFERENCES (ROUND-2)
Round 2 Progress on Erdős Problem 23
1) ROUND-2 OBJECTIVE
Path (A): proof track via sharper universal bounds and counterexample localization. We build on Round 1's elementary bounds to (i) obtain a strictly smaller unconditional constant than $25/8$, and (ii) constrain the edge-density regime in which a counterexample to $\tau_B(G)\le n^2$ could possibly occur.2) ROUND-1 FOUNDATION USED
We use the following Round 1 results as established inputs.- (R1--MaxCut) For any graph with $m$ edges, $\tau_B(G)\le m/2$.
- (R1--Mantel) If $G$ is triangle-free on $N$ vertices then $m\le N^2/4$.
- (R1--Sharpness) The balanced blow-up of $C_5$ on $5n$ vertices satisfies $\tau_B(G)=n^2$.
3) NEW INSIGHT / TOOL (ROUND-2)
Norin--Sun inequality.
For every graph $G$ on $N$ vertices, \[ \alpha_1(G)+\tau_B(G)\le \frac{N^2}{4}, \] where $\alpha_1(G)$ is the maximum size of a triangle-independent edge set. If $G$ is triangle-free then $\alpha_1(G)=|E(G)|=m$, hence \[ \tau_B(G)\le \frac{N^2}{4}-m. \]Balogh--Clemen--Lidický $2021$.
For all sufficiently large $N$, every triangle-free graph on $N$ vertices satisfies \[ \tau_B(G)\le \frac{N^2}{23.5}, \] and Erdős' conjectured bound $\tau_B(G)\le N^2/25$ holds outside a narrow intermediate edge-density window.4) ATTACK PLAN (ROUND-2)
Round 1 gave $\tau_B(G)\le m/2$ and $m\le N^2/4$, hence $\tau_B(G)\le N^2/8$. Round 2 adds the complementary bound $\tau_B(G)\le N^2/4-m$ (triangle-free case), so that taking the minimum improves the worst-case constant and forces any potential counterexample to lie in a mid-density regime.5) WORK (ROUND-2)
Let $G$ be triangle-free with $|V(G)|=N$ and $|E(G)|=m$.Theorem (Round-2 universal improvement): For triangle-free $G$ on $N$ vertices with $m$ edges, \[
\tau_B(G)\le \min\!\left(\frac{m}{2},\,\frac{N^2}{4}-m\right)\le \frac{N^2}{12}.
\] In particular, for $N=5n$, \[
\tau_B(G)\le \frac{25}{12}n^2.
\]
Proof: The inequality $\tau_B(G)\le m/2$ is (R1--MaxCut). The inequality $\tau_B(G)\le N^2/4-m$ follows from Norin--Sun and triangle-freeness. Thus $\tau_B(G)\le \min(m/2,\,N^2/4-m)$. The two expressions intersect at $m/2=N^2/4-m$, i.e. $m=N^2/6$, where the common value equals $N^2/12$. Therefore $\min(m/2,\,N^2/4-m)\le N^2/12$ for all $m$, proving the claim. Substituting $N=5n$ gives $\tau_B(G)\le 25n^2/12$. ∎
Corollary (Round-2 localization of potential counterexamples): Let $G$ be triangle-free on $5n$ vertices with $m$ edges. If $\tau_B(G)>n^2$, then \[
2n^2<m<\frac{21}{4}n^2.
\]
Proof: If $\tau_B(G)>n^2$, then Theorem Theorem (Round-2 universal improvement) implies both $m/2>n^2$ and $25n^2/4-m>n^2$, i.e. $m>2n^2$ and $m<21n^2/4$. ∎
Best known asymptotic constant on $5n$ vertices.
Balogh--Clemen--Lidický prove that for all sufficiently large $N$, $\tau_B(G)\le N^2/23.5$ for every triangle-free $G$ on $N$ vertices. For $N=5n$ this becomes \[ \tau_B(G)\le \frac{(5n)^2}{23.5}=\frac{50}{47}n^2\approx 1.064\,n^2 \qquad (n\ \text{sufficiently large}). \]6) ADVERSARIAL VERIFICATION (ROUND-2)
- The optimization in Theorem Theorem (Round-2 universal improvement) is exact: solving $m/2=N^2/4-m$ gives $m=N^2/6$ and bound $N^2/12$.
- Corollary Corollary (Round-2 localization of potential counterexamples) correctly substitutes $N=5n$ into $N^2/4$.
- All bounds are consistent with the balanced $C_5$ blow-up, which has $m=5n^2$ and $\tau_B(G)=n^2$.
7) FINAL (ROUND-2)
UNRESOLVED (but strictly advanced). Round 2 proves the improved unconditional bound $\tau_B(G)\le 25n^2/12$ for all triangle-free $G$ on $5n$ vertices, and localizes any hypothetical counterexample to the intermediate edge-count regime $2n^2<m<21n^2/4$.8) COMPLETION ESTIMATE (ROUND-2)
\[ \textbf{COMPLETION: }65\%. \]9) REFERENCES (ROUND-2)
S. Norin, Y. R. Sun, Triangle-independent sets vs. cuts, arXiv:1602.04370.J. Balogh, F. C. Clemen, B. Lidický, Max Cuts in Triangle-free Graphs, arXiv:2103.14179.