gpt pro 5.4 attempt on Math-926

Problem 766 (Round 2)

1) ROUND-2 OBJECTIVE

I pursue (C) obstruction/correction. The exact function $f(n;k,l)$ is still open, but Round 1 left its most important heuristic unproved: that the minimiser should be bipartite. The best Round-2 step is therefore to turn this heuristic into a theorem, thereby reducing the problem asymptotically to a finite family of bipartite graphs and solving the endpoint case $l=\lfloor k^2/4\rfloor$ exactly for large $n$.

2) Round-1 FOUNDATION USED

I use the following Round-1 facts.

• For $a:=\min\{t: l\le t(k-t)\}$ one has \[ f(n;k,l)\le \mathrm{ex}(n;K_{a,k-a})=O\left(n^{2-1/a}\right). \]
• Since $l\le \lfloor k^2/4\rfloor$, there exists at least one bipartite graph on $k$ vertices and $l$ edges.

Round 1 stated that one expects the minimiser to be bipartite; Round 2 proves that this is correct for all sufficiently large $n$.

3) NEW INSIGHT / TOOL (ROUND-2)

The new tool is the Erdős--Stone--Simonovits theorem. It implies that every fixed non-bipartite graph $H$ has \[ \mathrm{ex}(n;H)=\Theta(n^2). \] But Round 1 already supplied a bipartite $k$-vertex $l$-edge graph $B$ with \[ \mathrm{ex}(n;B)=O\left(n^{2-1/a}\right)=o(n^2). \] Therefore, for fixed $(k,l)$ in the range $k<l\le \lfloor k^2/4\rfloor$, every asymptotic minimiser for $f(n;k,l)$ must be bipartite.

This immediately yields an exact endpoint identification when $l=\lfloor k^2/4\rfloor$: among bipartite graphs on $k$ vertices, the only graph with that many edges is the balanced complete bipartite graph $K_{\lfloor k/2\rfloor,\lceil k/2\rceil}$.

4) ATTACK PLAN (ROUND-2)

The exact Round-1 gap was:

"The minimiser should be bipartite" was only heuristic.

I close that gap in three steps.

1. Use Erdős--Stone--Simonovits to show every non-bipartite candidate has quadratic extremal number.
2. Use the Round-1 Kővári--Sós--Turán upper bound to produce one bipartite candidate with subquadratic extremal number.
3. Because there are only finitely many $k$-vertex $l$-edge graphs, conclude that for sufficiently large $n$ the minimum is attained inside the bipartite subfamily.

Then I treat the endpoint $l=\lfloor k^2/4\rfloor$ by a direct uniqueness argument.

5) WORK (ROUND-2)

For fixed integers $k,l$ with $k<l\le \lfloor k^2/4\rfloor$, let \[ \mathcal G_{k,l}:=\{G: |V(G)|=k,\ |E(G)|=l\} \] and let $\mathcal B_{k,l}\subseteq \mathcal G_{k,l}$ be the bipartite members.

Lemma 5.1.

For every fixed non-bipartite graph $H$, \[ \mathrm{ex}(n;H)=\left(1-\frac{1}{\chi(H)-1}+o(1)\right)\frac{n^2}{2}. \] In particular, if $\chi(H)\ge 3$, then \[ \mathrm{ex}(n;H)=\Theta(n^2). \]

Justification. This is the Erdős--Stone--Simonovits theorem.

Lemma 5.2.

There exists $B\in \mathcal B_{k,l}$ such that \[ \mathrm{ex}(n;B)=o(n^2). \]

Proof. By the assumption $l\le \lfloor k^2/4\rfloor$, there exists a bipartite graph on $k$ vertices with $l$ edges. More concretely, by the Round-1 construction there is some $a\in\{1,\dots,\lfloor k/2\rfloor\}$ with $l\le a(k-a)$ and a graph $B\subseteq K_{a,k-a}$ having $k$ vertices and $l$ edges. Then \[ \mathrm{ex}(n;B)\le \mathrm{ex}(n;K_{a,k-a})=O\left(n^{2-1/a}\right)=o(n^2) \] by the Kővári--Sós--Turán theorem. $\square$

Theorem 5.3 (asymptotic bipartite reduction).

For every fixed $k,l$ with $k<l\le \lfloor k^2/4\rfloor$, there exists $N=N(k,l)$ such that for all $n\ge N$, \[ f(n;k,l)=\min\{\mathrm{ex}(n;B): B\in \mathcal B_{k,l}\}. \] In words: for sufficiently large $n$, the minimisation defining $f(n;k,l)$ may be restricted to bipartite graphs.

Proof. By definition, \[ f(n;k,l)=\min\{\mathrm{ex}(n;G): G\in \mathcal G_{k,l}\}. \] Let \[ M_B(n):=\min\{\mathrm{ex}(n;B): B\in \mathcal B_{k,l}\}. \] Clearly $f(n;k,l)\le M_B(n)$.

Now fix any non-bipartite graph $H\in \mathcal G_{k,l}\setminus \mathcal B_{k,l}$. By Lemma 5.1, \[ \mathrm{ex}(n;H)=\Theta(n^2). \] By Lemma 5.2, there exists some bipartite $B\in \mathcal B_{k,l}$ with \[ \mathrm{ex}(n;B)=o(n^2). \] Hence for each fixed non-bipartite $H$ there exists $N_H$ such that \[ \mathrm{ex}(n;B)<\mathrm{ex}(n;H)\qquad\text{for all }n\ge N_H. \] Since $\mathcal G_{k,l}\setminus \mathcal B_{k,l}$ is finite, taking \[ N:=\max_H N_H \] over all non-bipartite $H\in \mathcal G_{k,l}\setminus \mathcal B_{k,l}$ shows that for every $n\ge N$, every non-bipartite candidate has larger extremal number than some bipartite candidate. Therefore the minimum is attained within $\mathcal B_{k,l}$, i.e. \[ f(n;k,l)=M_B(n)=\min\{\mathrm{ex}(n;B): B\in \mathcal B_{k,l}\}. \] $\square$

Corollary 5.4.

For fixed $k,l$ in the stated range, \[ f(n;k,l)=o(n^2). \]

Proof. Immediate from Theorem 5.3 and Lemma 5.2. $\square$

Theorem 5.5 (endpoint $l=\lfloor k^2/4\rfloor$).

For fixed $k$ and sufficiently large $n$, \[ f\left(n;k,\lfloor k^2/4\rfloor\right)=\mathrm{ex}\left(n;K_{\lfloor k/2\rfloor,\lceil k/2\rceil}\right). \]

Proof. By Theorem 5.3, for sufficiently large $n$ the minimum may be taken over bipartite graphs with $k$ vertices and $\lfloor k^2/4\rfloor$ edges. But by Mantel's theorem, a bipartite graph on $k$ vertices has at most $\lfloor k^2/4\rfloor$ edges, with equality only for the balanced complete bipartite graph \[ K_{\lfloor k/2\rfloor,\lceil k/2\rceil}. \] Therefore the bipartite family $\mathcal B_{k,\lfloor k^2/4\rfloor}$ consists of this single graph, and the displayed identity follows. $\square$

What remains open after Round 2.

The strict monotonicity question is now sharply reduced: for fixed $k$ and large $n$, it is entirely a question about comparing extremal numbers of finitely many bipartite graphs on $k$ vertices. In particular, no non-bipartite graph can be asymptotically extremal in the minimisation problem.

6) ADVERSARIAL VERIFICATION

• Does Theorem 5.3 hold for all $n$? No. The argument is asymptotic, since it compares $\Theta(n^2)$ with $o(n^2)$. So the conclusion is correctly stated only for sufficiently large $n$.
• Could a non-bipartite graph have smaller extremal number than a bipartite one for small $n$? Yes, in principle. That is why the theorem includes the threshold $N(k,l)$.
• Why is finiteness important? Because the set of $k$-vertex $l$-edge graphs is finite, one can take the maximum of the finitely many thresholds $N_H$.
• Is the endpoint uniqueness claim correct? Yes. Among all bipartite graphs on $k$ vertices, the maximum possible number of edges is $\lfloor k^2/4\rfloor$, attained uniquely by the balanced complete bipartite graph.
• Did Round 2 solve monotonicity? No. It only localises the monotonicity problem to the finite bipartite family, which is a strict structural advance but not a full solution.

7) FINAL

UNRESOLVED (BUT STRICTLY ADVANCED). Round 2 proves the key structural correction that Round 1 only conjectured: for every fixed $k,l$ with $k<l\le \lfloor k^2/4\rfloor$, the minimiser in the definition of $f(n;k,l)$ is asymptotically bipartite. Moreover, \[ f\left(n;k,\lfloor k^2/4\rfloor\right)=\mathrm{ex}\left(n;K_{\lfloor k/2\rfloor,\lceil k/2\rceil}\right) \] for all sufficiently large $n$. This reduces the remaining difficulty to a finite comparison problem inside the bipartite regime.

8) COMPLETION ESTIMATE (MANDATORY)

COMPLETION: 35%

9) REFERENCES

• Erdős Problem #766, statement and historical note (official problem page).
• Erdős--Stone--Simonovits theorem.
• Kővári--Sós--Turán theorem.
• Mantel's theorem.

10) OUTPUT FORMAT

Here the Erdős problem number is $x=766$.