gpt pro 5.2 attempt on Math-858

1) FORMAL RESTATEMENT

Let $\mathcal{F}$ be a finite family of finite simple graphs. Define $\mathrm{ex}(n;\mathcal{F})$ to be the maximum number of edges in an $n$-vertex graph containing no subgraph isomorphic to any $F\in\mathcal{F}$.

Assume $\mathcal{F}$ contains at least one bipartite graph. The question is whether there always exists a bipartite graph $G\in\mathcal{F}$ and a constant $C_{\mathcal{F}}>0$ such that for all $n$, \[ \mathrm{ex}(n;G) \le C_{\mathcal{F}}\, \mathrm{ex}(n;\mathcal{F}). \] (Using Vinogradov notation: $\mathrm{ex}(n;G)\ll_{\mathcal{F}} \mathrm{ex}(n;\mathcal{F})$.)

2) QUICK LITERATURE/CONTEXT CHECK

The source file lists this as a problem of Erdős and Simonovits and points to related problem [180] and the graphs collection. The Erdős Problems website currently lists #575 as open (as of the access date shown there).

3) ATTACK PLAN

Proof-track ideas: - Identify natural sufficient conditions on $\mathcal{F}$ under which $\mathrm{ex}(n;\mathcal{F})$ is within a constant factor of $\min\{\mathrm{ex}(n;H):H\in\mathcal{F}\text{ bipartite}\}$. - Use subgraph containment relations inside $\mathcal{F}$: if one bipartite member is contained in all others, the problem becomes trivial.

Disproof-track ideas: - Search for a family $\mathcal{F}$ with multiple bipartite members such that avoiding all of them simultaneously forces a much smaller edge count than avoiding any one of them. - Attempt small $n$ brute force over families of small graphs to see whether a counterexample pattern emerges.

I provide two unconditional "sanity" lemmas (covering some degenerate and some structured cases) but do not resolve the general question.

4) WORK

FAST REALITY CHECK (sanity on degenerate cases). - If $K_2\in\mathcal{F}$ then $\mathrm{ex}(n;\mathcal{F})=0$ for all $n$, and choosing $G=K_2$ makes the desired inequality trivial. - If $\mathcal{F}$ contains a bipartite graph $G$ that is a subgraph of every $F\in\mathcal{F}$, then forbidding $\mathcal{F}$ is the same as forbidding $G$.

These are formalized below.

Lemma 575.1 (trivial case when $K_2\in\mathcal{F}$). If $K_2\in\mathcal{F}$ then $\mathrm{ex}(n;\mathcal{F})=0$ and taking $G=K_2$ satisfies $\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})$. Proof. A graph contains no $K_2$ subgraph if and only if it has no edges. So forbidding $K_2$ forces $e(G)=0$, hence $\mathrm{ex}(n;\mathcal{F})=0$. Also $\mathrm{ex}(n;K_2)=0$. Thus $\mathrm{ex}(n;K_2)=0\le C\cdot 0$ for any constant $C$. $\square$

Lemma 575.2 (subgraph-dominating bipartite member). Suppose there exists a bipartite graph $G\in\mathcal{F}$ such that $G$ is a subgraph of every $F\in\mathcal{F}$. Then for every $n$, \[ \mathrm{ex}(n;\mathcal{F})=\mathrm{ex}(n;G). \] In particular, the conjectured inequality holds with this $G$ and constant $C_{\mathcal{F}}=1$. Proof. Let $H$ be any $n$-vertex graph. If $H$ contains some $F\in\mathcal{F}$, then (since $G\subseteq F$) the same copy of $F$ contains a copy of $G$; hence $H$ contains $G$. Therefore, if $H$ is $G$-free then $H$ is $F$-free for every $F\in\mathcal{F}$, i.e. $H$ is $\mathcal{F}$-free. This shows that the class of $G$-free graphs is a subset of the class of $\mathcal{F}$-free graphs, so $\mathrm{ex}(n;G)\le \mathrm{ex}(n;\mathcal{F})$.

Conversely, because $G\in\mathcal{F}$, any $\mathcal{F}$-free graph is in particular $G$-free. So the class of $\mathcal{F}$-free graphs is a subset of the class of $G$-free graphs, giving $\mathrm{ex}(n;\mathcal{F})\le \mathrm{ex}(n;G)$.

Combining the two inequalities yields equality. $\square$

Why the general case is hard. Outside Lemma 575.2, forbidding multiple graphs can interact nontrivially: graphs that are extremal for one forbidden subgraph can be structurally very different from extremal graphs for another. The question asks whether this interaction can ever reduce $\mathrm{ex}(n;\mathcal{F})$ by more than a constant factor compared to every bipartite member of $\mathcal{F}$.

5) VERIFICATION

- Lemma 575.1 and Lemma 575.2 follow directly from the definition of subgraph containment and the monotonicity $\mathrm{ex}(n;\mathcal{F})\le \mathrm{ex}(n;F)$ for each $F\in\mathcal{F}$. - In Lemma 575.2, both inclusions of forbidden classes were checked carefully: (i) $G$-free implies $\mathcal{F}$-free by $G\subseteq F$ for all $F$, and (ii) $\mathcal{F}$-free implies $G$-free because $G\in\mathcal{F}$.

6) FINAL

UNRESOLVED

(i) Strongest fully proved partial result: The conjecture holds in the degenerate case $K_2\in\mathcal{F}$ (Lemma 575.1) and in the structured case where some bipartite $G\in\mathcal{F}$ is a subgraph of every member of $\mathcal{F}$ (Lemma 575.2).

(ii) First gap (crisp): Either prove that for every finite family $\mathcal{F}$ containing a bipartite graph there exists a bipartite $G\in\mathcal{F}$ with $\mathrm{ex}(n;G)=O_{\mathcal{F}}(\mathrm{ex}(n;\mathcal{F}))$, or exhibit a concrete family $\mathcal{F}$ for which \[\frac{\mathrm{ex}(n;G)}{\mathrm{ex}(n;\mathcal{F})}\to\infty\quad\text{for every bipartite }G\in\mathcal{F}. \]

(iii) Top 3 next moves (concrete): 1. Systematically search for a counterexample among small families $\mathcal{F}$ built from standard bipartite obstructions (even cycles, complete bipartite graphs, trees) by comparing growth exponents. 2. Prove a "winner-takes-all" principle for families whose bipartite members have distinct known extremal exponents: show $\mathrm{ex}(n;\mathcal{F})$ is within a constant of the smallest exponent member. 3. Try to construct $\mathcal{F}=\{G_1,G_2\}$ with incomparable bipartite graphs where known extremal constructions for $G_1$ necessarily contain $G_2$ and vice versa, potentially forcing a new (smaller) extremal order.

(iv) Minimal counterexample structure: A counterexample would require a family $\mathcal{F}$ with at least two bipartite members such that avoiding all of them forces a substantially sparser graph than avoiding any one of them (ratio diverging with $n$). This would likely involve two bipartite graphs with extremal constructions of very different "geometry" so that the intersection of the two avoidance properties severely restricts edge density.