GPT 6 Astra Ultra attempt on Math-5995
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Erdős Problem #924: an edge-Ramsey graph with no larger clique
Statement and result
For every $k\geq2$ and $\ell\geq3$, there is a $K_{\ell+1}$-free graph whose every $k$-colouring of the edges contains a monochromatic $K_\ell$. We reconstruct a partite construction. The external inputs are the finite Ramsey theorem and the finite Hales--Jewett theorem; the graph construction, embeddings, and clique preservation are proved below.The one word theorem used
The finite Hales--Jewett theorem says that for any finite nonempty alphabet $\Sigma$ and positive integer $k$, some integer $d$ has this property: every $k$-colouring of $\Sigma^d$ admits a monochromatic combinatorial line. Such a line is specified by a nonempty set $J\subseteq[d]$ of variable positions and fixed letters outside $J$; its words put the same letter $\sigma\in\Sigma$ at every position of $J$, as $\sigma$ varies. This theorem is used as an external input, not reproved.An induced bipartite partite lemma
Let $B$ be a bipartite graph with specified parts $U,V$, containing at least one edge. There is a bipartite graph $C$, with specified parts, such that every $k$-edge-colouring of $C$ has a monochromatic induced copy of $B$ respecting the two parts.Use $\Sigma=E(B)$, where an edge is written as its ordered pair $(u,v)\in U\times V$, and choose $d$ by Hales--Jewett. Give $C$ parts $U^d,V^d$, joining $\mathbf u$ to $\mathbf v$ exactly when $u_iv_i\in E(B)$ for every coordinate $i$. Edges of $C$ are in bijection with words of $\Sigma^d$. A monochromatic line with variable positions $J$ and fixed edges $(u_i,v_i)$ gives maps \[ u\longmapsto\mathbf u(u),\quad \mathbf u(u)_i=\begin{cases}u&i\in J,\\u_i&i\notin J,\end{cases} \qquad v\longmapsto\mathbf v(v),\quad \mathbf v(v)_i=\begin{cases}v&i\in J,\\v_i&i\notin J.\end{cases} \] The nonempty variable set makes both maps injective. Their images are adjacent exactly when $uv\in E(B)$, so this is an induced part-respecting copy, including any isolated vertices of $B$. Its edges are precisely the line's edges and hence all have one colour.
Making one pair of parts monochromatic
Suppose $F$ is a graph with specified independent parts $V_1,\ldots,V_q$, and fix a pair $a<b$ for which the bipartite slice $B=F[V_a\cup V_b]$ has an edge. Construct $C$ for this slice by the preceding lemma, using its parts as parts $a,b$. For every induced part-respecting embedding of $B$ into $C$, glue a fresh copy of $F$ along exactly that embedded slice. All its other vertices are new and private to that copy, placed in their original labelled parts; give no edges between private vertices of different copies.The resulting graph $F^+$ is finite and $q$-partite. Every edge-colouring of $F^+$ restricts to one on $C$, hence supplies a monochromatic embedded $B$. The corresponding attached copy of $F$ has all edges between its parts $a,b$ of one colour.
Moreover, \[ \omega(F^+)\leq\max\{2,\omega(F)\}. \tag{1} \] A clique entirely in $C$ has at most two vertices. Any clique containing a private vertex must lie in that vertex's attached copy: it has no neighbours among other copies' private vertices, and its neighbours in $C$ belong to its own glued slice. Since that slice was embedded induced, the edges in the attached copy are exactly those of $F$. This proves $1$. If a slice has no edges, the step can simply be omitted.
Iterating over all pairs
Let $q$ be a finite $k$-colour Ramsey number for $K_\ell$. Build a $q$-partite graph $F_0$ by taking one disjoint copy of $K_\ell$ for each $\ell$-subset of $[q]$, assigning its vertices to those labelled parts. Then $\omega(F_0)=\ell$. In particular every pair of parts has at least one edge.List the $\binom q2$ pairs of part labels. Starting from $F_0$, perform the preceding construction for each pair, obtaining $F_1,\ldots,F_{\binom q2}$. By $1$, every one of these graphs has clique number at most $\ell$.
Now colour the edges of the last graph arbitrarily. The final construction step gives a part-respecting copy of the preceding graph on which the final pair of parts is monochromatic. Inside that copy apply the preceding step, and continue backwards. Restriction to a smaller part-respecting copy preserves every monochromatic-pair property already acquired. We obtain a copy of $F_0$ such that, for each pair of labels $a,b$, all its edges between those parts share a colour $c(a,b)$.
These colours define an edge-colouring of the complete graph on $[q]$. By the choice of $q$, some $\ell$-subset $L\subseteq[q]$ has all its pair colours equal. The designated $K_\ell$ in $F_0$ for that very label set $L$ is then monochromatic. Thus the last constructed graph has the required edge-Ramsey property and contains no $K_{\ell+1}$.