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Math-5376-A5 · revision r1 · on Math-5376

gpt pro 5.2 attempt on Math-5376, version 3

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Contents
  1. Round 2 Progress on Erdős Problem 23
  2. 1) ROUND-2 OBJECTIVE
  3. 2) ROUND-1 FOUNDATION USED
  4. 3) NEW INSIGHT / TOOL (ROUND-2)
  5. Norin--Sun inequality.
  6. Balogh--Clemen--Lidický $2021$.
  7. 4) ATTACK PLAN (ROUND-2)
  8. 5) WORK (ROUND-2)
  9. Best known asymptotic constant on $5n$ vertices.
  10. 6) ADVERSARIAL VERIFICATION (ROUND-2)
  11. 7) FINAL (ROUND-2)
  12. 8) COMPLETION ESTIMATE (ROUND-2)
  13. 9) REFERENCES (ROUND-2)
  14. Round 3 Progress on Erdős Problem 23
  15. 1) ROUND-3 OBJECTIVE
  16. 2) ROUND-1/2 FOUNDATION USED
  17. 3) NEW INSIGHT / TOOL (ROUND-3)
  18. $C_5$-colorability and "one monochromatic edge" cuts.
  19. 4) ATTACK PLAN (ROUND-3)
  20. 5) WORK (ROUND-3)
  21. 5.1 A five-part AM--GM lemma
  22. 5.2 The conjectured $n^2$ bound for $C_5$-colorable graphs
  23. 6) ADVERSARIAL VERIFICATION (ROUND-3)
  24. 7) FINAL (ROUND-3)
  25. 8) COMPLETION ESTIMATE (ROUND-3)
  26. 9) REFERENCES (ROUND-3)

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)

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.

Round 3 Progress on Erdős Problem 23

1) ROUND-3 OBJECTIVE

Path (A): prove the conjectured $n^2$ bound on a large structural subclass. The full statement remains open, but we prove \[ \tau_B(G)\le n^2 \] exactly for every triangle-free graph $G$ on $5n$ vertices that admits a homomorphism to $C_5$ (equivalently, every subgraph of a $C_5$ blow-up). This yields a rigorous reduction: any counterexample must be non-$C_5$-colorable.

2) ROUND-1/2 FOUNDATION USED

We use the definition of $\tau_B(G)$ and the Round 1 sharpness example (balanced $C_5$ blow-up). Round 2 bounds provide context but are not needed for the main new theorem.

3) NEW INSIGHT / TOOL (ROUND-3)

$C_5$-colorability and "one monochromatic edge" cuts.
A 2-coloring of the vertices of $C_5$ makes exactly one cycle-edge monochromatic. If $G$ maps homomorphically to $C_5$, then pulling back such a 2-coloring gives a cut in $G$ whose deleted edges lie entirely in one consecutive pair of color classes. Thus $\tau_B(G)$ is controlled by the smallest consecutive cross-edge set.

4) ATTACK PLAN (ROUND-3)

We prove:
If $G$ admits a homomorphism $G\to C_5$ and $|V(G)|=5n$, then $\tau_B(G)\le n^2$. Moreover, equality forces $G$ to be the balanced complete blow-up of $C_5$.
The key quantitative input is an AM--GM lemma showing that among five nonnegative numbers summing to $5n$, some adjacent product is at most $n^2$.

5) WORK (ROUND-3)

5.1 A five-part AM--GM lemma

Lemma: Let $a_1,\dots,a_5\ge 0$ with $\sum_{i=1}^5 a_i=5n$. Then \[ \min_{i\in \mathbb Z/5\mathbb Z} a_i a_{i+1} \le n^2. \] Moreover, $\min_i a_i a_{i+1}=n^2$ if and only if $a_1=\cdots=a_5=n$.
Proof: If some $a_i=0$ the claim is immediate, so assume $a_i>0$ for all $i$. Suppose for contradiction that $a_i a_{i+1}>n^2$ for all $i$. Multiplying these five inequalities yields \[ \prod_{i=1}^5 (a_i a_{i+1})>n^{10}. \] But $\prod_{i=1}^5 (a_i a_{i+1})=(a_1a_2a_3a_4a_5)^2$, so $a_1\cdots a_5>n^5$. By AM--GM, \[ a_1\cdots a_5 \le \left(\frac{a_1+\cdots+a_5}{5}\right)^5=n^5, \] a contradiction. Hence $\min_i a_i a_{i+1}\le n^2$.

For the equality statement, assume $\min_i a_i a_{i+1}=n^2$. Then all $a_i a_{i+1}\ge n^2$, so $(a_1\cdots a_5)^2=\prod_i (a_i a_{i+1})\ge n^{10}$ and thus $a_1\cdots a_5\ge n^5$. AM--GM gives $a_1\cdots a_5\le n^5$, so equality holds in AM--GM, forcing $a_1=\cdots=a_5=n$. ∎

5.2 The conjectured $n^2$ bound for $C_5$-colorable graphs

Definition: A graph $G$ is $C_5$-colorable if there exists a homomorphism $\varphi:V(G)\to V(C_5)=\{1,2,3,4,5\}$ such that for every edge $uv\in E(G)$, $\varphi(u)\varphi(v)$ is an edge of $C_5$. Equivalently, writing $V_i:=\varphi^{-1}(i)$, every edge of $G$ lies between $V_i$ and $V_{i\pm 1}$ (indices mod $5$).
Theorem: Let $G$ be $C_5$-colorable with $|V(G)|=5n$. Then \[ \tau_B(G)\le n^2. \] Moreover, if $\tau_B(G)=n^2$, then $|V_1|=\cdots=|V_5|=n$ and $e(V_i,V_{i+1})=n^2$ for all $i$; in particular, $G$ is the balanced complete blow-up of $C_5$.
Proof: Let $\varphi:V(G)\to \{1,2,3,4,5\}$ be a homomorphism to $C_5$ and let $V_i:=\varphi^{-1}(i)$. Then $V(G)=V_1\cup\cdots\cup V_5$ and every edge of $G$ lies between $V_i$ and $V_{i\pm 1}$. (Consequently $G$ is triangle-free, since a triangle would map to a triangle in $C_5$.)

Fix $i$ (indices mod $5$). There exists a 2-coloring of $V(C_5)$ in which exactly the edge $i(i+1)$ is monochromatic and the other four cycle-edges are bichromatic. Pulling back this 2-coloring via $\varphi$ yields a bipartition $V(G)=X\cup Y$ with the property that every edge of $G$ except those between $V_i$ and $V_{i+1}$ crosses the cut. Hence deleting the (possibly all) edges between $V_i$ and $V_{i+1}$ makes $G$ bipartite, so \[ \tau_B(G)\le e(V_i,V_{i+1})\qquad \text{for each } i, \] and therefore \[ \tau_B(G)\le \min_i e(V_i,V_{i+1}) \le \min_i |V_i||V_{i+1}|. \] Since $\sum_i |V_i|=5n$, Lemma Lemma implies $\min_i |V_i||V_{i+1}|\le n^2$, proving $\tau_B(G)\le n^2$.

Assume now $\tau_B(G)=n^2$. Then \[ n^2=\tau_B(G)\le \min_i e(V_i,V_{i+1})\le \min_i |V_i||V_{i+1}|\le n^2, \] so all inequalities are equalities. In particular $\min_i |V_i||V_{i+1}|=n^2$, so Lemma Lemma forces $|V_1|=\cdots=|V_5|=n$. Then each $e(V_i,V_{i+1})\le |V_i||V_{i+1}|=n^2$, and since also $\min_i e(V_i,V_{i+1})=n^2$, we must have $e(V_i,V_{i+1})=n^2$ for every $i$. Thus each consecutive pair forms $K_{n,n}$ and $G$ is the balanced complete blow-up of $C_5$. ∎

6) ADVERSARIAL VERIFICATION (ROUND-3)

  • Edge cases: If some $V_i=\emptyset$, then $e(V_i,V_{i+1})=0$ and the cut above deletes $0$ edges, giving $\tau_B(G)=0\le n^2$.
  • Tightness: The balanced complete $C_5$ blow-up attains $\tau_B(G)=n^2$.
  • No hidden assumptions: The argument uses only the existence of a homomorphism $G\to C_5$ and the identity that a cut of $C_5$ leaves one monochromatic edge.

7) FINAL (ROUND-3)

UNRESOLVED (but strictly advanced). We prove Erdős' bound $\tau_B(G)\le n^2$ for the entire subclass of $C_5$-colorable graphs on $5n$ vertices, with a rigidity statement characterizing equality. Consequently, any counterexample to the full conjecture must be non-$C_5$-colorable.

8) COMPLETION ESTIMATE (ROUND-3)

\[ \textbf{COMPLETION: }72\%. \]

9) REFERENCES (ROUND-3)

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.