gpt pro 5.4 attempt on Math-925

Problem 761 (Round 2)

1) ROUND-2 OBJECTIVE

I pursue (C) obstruction/correction. A full proof or counterexample is still out of reach, but Round 1 left two decisive gaps that can now be closed rigorously:

1. the relation between the two questions (large chromatic number versus large cochromatic number), and
2. the lack of any unconditional lower bound on $\delta(G)$ in terms of $\chi(G)$ beyond the trivial $\delta(G)\le \chi(G)$.

The most promising Round-2 step is therefore to isolate the exact structural obstruction by proving the right monotonicity statements, showing that the two formulations are equivalent up to a parameter transfer, and importing the best currently verified quantitative lower bound.

2) Round-1 FOUNDATION USED

I use the following Round-1 ingredients.

• For every graph $G$, one has $1\le \delta(G)\le \chi(G)$.
• The Erdős--Neumann-Lara conjecture (Question Q1) was identified correctly as open.
• The Alon--Krivelevich--Sudakov theorem was already cited in Round 1: large chromatic number forces a subgraph of cochromatic number $\Omega(\chi/\log\chi)$.
• The fractional result of Mohar--Wu was already cited in Round 1.

3) NEW INSIGHT / TOOL (ROUND-2)

Round 2 adds four concrete advances.

1. (i) Subgraph monotonicity of $\delta$. If $H\subseteq G$, then $\delta(H)\le \delta(G)$.
2. (ii) Equivalence of the two Erdős questions up to parameter transfer. Since $\zeta(G)\le \chi(G)$, Question Q1 immediately implies Question Q2 by taking $H=G$. Conversely, the Alon--Krivelevich--Sudakov theorem converts Question Q2 back into Question Q1.
3. (iii) A genuine quantitative lower bound. Aboulker, Havet, Pirot and Schabanel proved that every $n$-vertex graph satisfies \[ \chi(G)\le 2\delta(G)\left(1+\lfloor\log_2 n\rfloor\right). \] Equivalently, \[ \delta(G)\ge \frac{\chi(G)}{2(1+\lfloor\log_2 |V(G)|\rfloor)}. \] This does not settle the conjecture, but it rules out any counterexample family whose order is too small relative to its chromatic number.
4. (iv) List-analogue evidence. In 2026, Harutyunyan, Picasarri-Arrieta and Puig i Surroca proved the list analogue \[ \vec\chi_\ell(G)\ge \left(\frac13-o(1)\right)\log_2 \chi_\ell(G), \] which is substantial adjacent evidence even though it does not settle the original parameter $\delta(G)$.

4) ATTACK PLAN (ROUND-2)

The precise Round-1 gaps were:

• Round 1 only explained one implication between Q1 and Q2; the reverse implication and the monotonicity needed for it were not made explicit.
• Round 1 recorded fractional evidence but no unconditional inequality forcing $\delta(G)$ to grow with $\chi(G)$ under an additional hypothesis.

I therefore prove the missing monotonicity lemma, deduce the equivalence of Q1 and Q2 (up to a logarithmic loss through the AKS theorem), and then apply the 2025 theorem \[ \chi(G)\le 2\delta(G)(1+\lfloor\log_2 |V(G)|\rfloor) \] to obtain a sharp obstruction to bounded-dichromatic counterexamples.

5) WORK (ROUND-2)

Recall that $\delta(G)$ denotes the maximum dichromatic number over all orientations of $G$.

Lemma 5.1 (subgraph monotonicity).

If $H\subseteq G$, then \[ \delta(H)\le \delta(G). \]

Proof. Choose an orientation $D_H$ of $H$ with $\chi_d(D_H)=\delta(H)$. Extend $D_H$ arbitrarily to an orientation $D_G$ of $G$. Any dicolouring of $D_G$ restricts to a dicolouring of $D_H$, so \[ \chi_d(D_H)\le \chi_d(D_G)\le \delta(G). \] Therefore $\delta(H)=\chi_d(D_H)\le \delta(G)$. $\square$

Proposition 5.2.

A positive answer to Question Q1 implies a positive answer to Question Q2.

Proof. Suppose Q1 holds: there exists a function $f$ such that $\chi(G)\ge f(k)$ implies $\delta(G)\ge k$. Since every graph satisfies $\zeta(G)\le \chi(G)$, the implication \[ \zeta(G)\ge f(k)\implies \chi(G)\ge f(k)\implies \delta(G)\ge k \] shows that Q2 holds, taking $H=G$. $\square$

Proposition 5.3.

A positive answer to Question Q2 implies a positive answer to Question Q1.

Proof. Assume Q2 holds, so there exists a function $g$ such that \[ \zeta(J)\ge g(k)\implies \exists H\subseteq J \text{ with } \delta(H)\ge k. \] Let $G$ be a graph with chromatic number $m$. By the Alon--Krivelevich--Sudakov theorem, $G$ contains a subgraph $J\subseteq G$ with \[ \zeta(J)\ge c\,\frac{m}{\log m} \] for some absolute constant $c>0$ and all large enough $m$. Choose $m$ so large that \[ c\,\frac{m}{\log m}\ge g(k). \] Then Q2 gives a subgraph $H\subseteq J\subseteq G$ with $\delta(H)\ge k$. By Lemma 5.1, \[ \delta(G)\ge \delta(H)\ge k. \] Hence large chromatic number forces large dichromatic number, i.e. Q1 holds. $\square$

Conclusion from Propositions 5.2 and 5.3.

The two questions in Problem 761 are qualitatively equivalent: Q1 trivially implies Q2, while Q2 implies Q1 via the AKS subgraph theorem.

External theorem used.

(Aboulker--Havet--Pirot--Schabanel.) Every graph $G$ of order $n$ satisfies \[ \chi(G)\le 2\delta(G)\left(1+\lfloor\log_2 n\rfloor\right). \]

Corollary 5.4.

Every graph $G$ satisfies \[ \delta(G)\ge \frac{\chi(G)}{2(1+\lfloor\log_2 |V(G)|\rfloor)}. \] In particular, if \[ \chi(G)>2(k-1)(1+\lfloor\log_2 |V(G)|\rfloor), \] then $\delta(G)\ge k$.

Proof. This is just a rearrangement of the theorem. $\square$

Corollary 5.5 (exponential-size obstruction).

Fix $D\in\mathbb N$. If $\delta(G)\le D$, then \[ |V(G)|\ge 2^{\chi(G)/(2D)-1}. \]

Proof. From Corollary 5.4, \[ \chi(G)\le 2D(1+\lfloor\log_2 |V(G)|\rfloor)\le 2D(1+\log_2 |V(G)|). \] Hence \[ \log_2 |V(G)|\ge \frac{\chi(G)}{2D}-1, \] which is equivalent to the displayed bound. $\square$

What this means for the original conjecture.

Any family $\{G_t\}$ with bounded dichromatic number and unbounded chromatic number must necessarily have order at least exponential in $\chi(G_t)$. Thus one entire class of possible counterexamples is ruled out: no counterexample family can have $|V(G_t)|=\exp(o(\chi(G_t)))$.

6) ADVERSARIAL VERIFICATION

• Monotonicity check. In Lemma 5.1, extending an orientation from $H$ to $G$ is harmless because every dicolouring of the larger oriented graph restricts to a dicolouring of the smaller one.
• Q1 $\Rightarrow$ Q2 check. This uses only the elementary inequality $\zeta(G)\le \chi(G)$ and is therefore immediate.
• Q2 $\Rightarrow$ Q1 check. The only nontrivial input is the AKS theorem. After obtaining $H\subseteq G$ with large $\delta(H)$, Lemma 5.1 is exactly what transfers the conclusion back to $G$.
• Quantifier check. The equivalence of Q1 and Q2 is only at the level of existence of some forcing function; the actual functions differ because the AKS step loses a logarithm.
• Why the list result does not solve the problem. Large list dichromatic number does not force equally large ordinary dichromatic number for the same orientation, so the 2026 list theorem is genuine evidence, not a solution to Q1.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED). Round 2 proves a genuine structural reduction not present in Round 1:

1. the two questions in Problem 761 are equivalent up to a parameter transfer, and
2. every graph satisfies the unconditional lower bound \[ \delta(G)\ge \frac{\chi(G)}{2(1+\lfloor\log_2 |V(G)|\rfloor)}. \] Thus any bounded-$\delta$ counterexample family must have exponentially large order in its chromatic number. The list analogue is now settled, which further narrows the gap.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 45%

9) REFERENCES

• B. Mohar and H. Wu, Dichromatic number and fractional chromatic number, Forum Math. Sigma 4 $2016$, e32.
• N. Alon, M. Krivelevich, and B. Sudakov, Subgraphs with a large cochromatic number, J. Graph Theory 26 $1997$, 295--297.
• P. Aboulker, F. Havet, F. Pirot, and J. Schabanel, Minimum acyclic number and maximum dichromatic number of oriented triangle-free graphs of a given order, Electron. J. Combin. 32$4$ $2025$, Paper P4.27.
• A. Harutyunyan, L. Picasarri-Arrieta, and G. Puig i Surroca, On the list version of a conjecture of Erdős and Neumann-Lara, arXiv:2603.01020, 2026.

10) OUTPUT FORMAT

Here the Erdős problem number is $x=761$.