GPT 6 Astra Ultra attempt on Math-5740
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Erdős Problem 565
Statement and scope.
The induced Ramsey number $R^*(G)$ is the least order of a graph $H$ whose every red-blue edge colouring contains a monochromatic copy of $G$ induced in $H$. Must $R^*(G)\leq2^{O(n)}$ hold for every $n$-vertex graph $G$? The general affirmative theorem is known. This attempt proves the exponential scale is necessary, gives a full exponential upper bound for complete targets, and supplies small induced hosts for stars and edgeless targets. The general construction is unresolved here.Complete targets reduce exactly to ordinary Ramsey numbers.
Write $R(n,n)$ for the ordinary two-colour clique Ramsey number. The complete host $K_{R(n,n)}$ shows $R^*(K_n)\leq R(n,n)$: any monochromatic clique is automatically induced as an uncoloured clique.Conversely, if $N<R(n,n)$, choose a colouring of $K_N$ with no monochromatic $K_n$. Restrict it to the edge set of any $N$-vertex host $H$. This restriction still has no monochromatic $K_n$, so no such host is induced Ramsey for $K_n$. Thus \[ R^*(K_n)=R(n,n). \tag{1} \] This argument checks the quantifier over all possible host graphs.
The elementary recurrence $R(a,b)\leq R(a-1,b)+R(a,b-1)$ follows by splitting the neighbours of one vertex according to edge colour. With the base cases $R(1,b)=R(a,1)=1$, induction using Pascal's identity gives \[ R(a,b)\leq\binom{a+b-2}{a-1}. \] Consequently \[ R^*(K_n)\leq\binom{2n-2}{n-1}\leq4^{n-1}. \tag{2} \]
A self-contained exponential lower bound.
Colour each edge of $K_N$ independently red or blue with equal probability. A specified $n$-set is monochromatic with probability $2^{1-\binom n2}$. Therefore the expected number of monochromatic $K_n$'s is \[ \binom Nn2^{1-\binom n2}. \] For $N=\lfloor2^{n/2}\rfloor$ this is at most $2^{1+n/2}/n!<1$ for every $n\geq4$. The last inequality holds at $n=4$, and its left-to-right ratio decreases when $n$ is increased because the factorial gains a factor $n+1>\sqrt2$. Some colouring therefore contains no monochromatic $K_n$. By $1$, \[ R^*(K_n)>\lfloor2^{n/2}\rfloor. \tag{3} \] Thus any bound valid for every $n$-vertex target must allow exponential growth. A polynomial general upper bound is impossible.Two other target families with induced hosts.
For the star on $n\geq2$ vertices, use a host star with $2n-3$ leaves. At least $n-1$ edges at its centre have the same colour. Their endpoints induce exactly the desired star, because the host has no edges between leaves. Hence \[ R^*(K_{1,n-1})\leq2n-2. \tag{4} \] For the edgeless $n$-vertex graph, the edgeless host of the same order works, with monochromaticity vacuous; fewer than $n$ vertices cannot contain the target. Its induced Ramsey number is therefore exactly $n$.The obstruction to extending these arguments.
For a general non-complete target, the complete host used in $2$ does not work: it contains no induced copy of any graph with a missing edge. The host must simultaneously retain the prescribed nonedges and force all its required edges into one colour. Equations $1$--$4$ do not provide such a construction for arbitrary targets, and so do not prove the main assertion.The full theorem of Aragão, Campos, Dahia, Filipe and Marciano gives $R_{\mathrm{ind}}(G;r)\leq r^{Crn}$ with an absolute constant $C$; setting $r=2$ settles the question. Its general embedding argument is not reconstructed or used to fill this attempt's gap. Primary source: https://arxiv.org/abs/2509.22629. Current statement: https://www.erdosproblems.com/565, checked 24 September 2026. The elementary results above are standard; no novelty is claimed.