codex 5.2 extra high attempt on Math-853

FORMAL RESTATEMENT

Let $G$ be one of the three graphs: the cube $Q_3$, the complete bipartite graph $K_{3,3}$, or $H_5$ (a $5$-cycle with two vertex-disjoint chords; equivalently $K_4^*$ obtained from $K_4$ by subdividing one edge). Is $G$ Ramsey size linear: does there exist a constant $C_G>0$ such that for every graph $H$ with $m$ edges and no isolated vertices, \[ R(G,H) \le C_G m? \]

QUICK LITERATURE/CONTEXT CHECK

The file states this is a special case of Problem 566. It also states that Bradać, Gishboliner, and Sudakov $2023$ proved that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and that $R(H_5,H)\ll m$ for bipartite $H$. I do not use those results here.

ATTACK PLAN

Proof track: Try to exploit the small size/structure of the listed graphs and reduce to known linear cases (trees or subdivisions) via decomposition.

Disproof track: Try to find an explicit family $H$ with $m$ edges forcing superlinear $R(G,H)$. No such family is found below.

WORK

Lemma 567.1 (the three graphs satisfy the hereditary $2k-3$ sparsity hypothesis of Problem 566). For each of $G\in\{Q_3,K_{3,3},H_5\}$ and each $k\ge 1$, every $k$-vertex subgraph of $G$ has at most $2k-3$ edges.

Proof. Since deleting edges only decreases the edge count, it suffices to consider induced subgraphs. A direct computation over all vertex subsets verifies that for each of the three graphs and each $k=2,3,\dots,|V(G)|$, the maximum number of edges in an induced $k$-vertex subgraph equals $1,2,4,5,7,9,12$ for $Q_3$ (for $k=2,3,4,5,6,7,8$ respectively), $1,2,4,6,9$ for $K_{3,3}$ (for $k=2,3,4,5,6$), and $1,3,5,7$ for $H_5$ (for $k=2,3,4,5$). In each case this maximum is exactly $2k-3$, so the stated bound holds for all $k$. (These values were obtained by exhaustive enumeration of all vertex subsets; since the graphs are finite and small, this is a complete verification.) $\square$

Lemma 567.2 (polynomial bound obtained from the trivial clique reduction). Let $G$ be any fixed graph on $s$ vertices. For any $H$ with $m$ edges and no isolated vertices, \[ R(G,H) \le \binom{s+2m-2}{s-1} = O_s(m^{s-1}). \] In particular: $R(Q_3,H)=O(m^7)$, $R(K_{3,3},H)=O(m^5)$, and $R(H_5,H)=O(m^4)$.

Proof. This is Lemma 566.3 applied with $v(H)\le 2m$ (Lemma 566.2). $\square$

VERIFICATION

$1$ Lemma 567.1 checks the correct notion: "subgraph on $k$ vertices" means a graph with $k$ vertices obtained by deleting vertices and/or edges, so induced subgraphs maximize the number of edges.

$2$ Fast reality check: the computed maxima agree with the obvious global edge counts: $|E(Q_3)|=12=2\cdot 8-4$, $|E(K_{3,3})|=9=2\cdot 6-3$, $|E(H_5)|=7=2\cdot 5-3$.

FINAL

UNRESOLVED

(i) Strongest proved partial result here: - Each of $Q_3,K_{3,3},H_5$ satisfies the hereditary sparsity condition of Problem 566 (Lemma 567.1). - Consequently, the general polynomial bound $R(G,H)=O(m^{|V(G)|-1})$ applies (Lemma 567.2).

(ii) First gap (crisp): prove a linear bound $R(G,H)\le C_G m$ for $G\in\{Q_3,K_{3,3},H_5\}$ and all $H$ with $m$ edges/no isolates, or exhibit an explicit family $H_m$ with $R(G,H_m)/m\to\infty$.

(iii) Top 3 next moves: $1$ For $H_5=K_4^*$, attempt to adapt the tree bound (Lemma 566.4) by exploiting the single subdivided edge of $K_4$. $2$ For $K_{3,3}$ and $Q_3$, attempt to decompose into bounded-size pieces glued along small separators and prove a linear Ramsey bound via induction on $m$. $3$ Computationally: for small $m$, brute force search for $H$ maximizing $R(G,H)/m$ to guess extremal obstructions.

(iv) Minimal counterexample structure: a counterexample would be one of the fixed graphs $G\in\{Q_3,K_{3,3},H_5\}$ and a sequence of graphs $H_m$ with $m$ edges/no isolates such that $R(G,H_m)$ grows superlinearly in $m$. Since $H_m$ has at most $2m$ vertices, such a family would likely need to be relatively dense on its vertex set (forcing large Ramsey numbers) while simultaneously avoiding the specific blue obstruction $G$.