Ramsey sequences with bounded clique size

A sequence of graphs $\{G_k\}$ is a Ramsey sequence if for every positive integer $k$, the graph $G_k$ is a proper subgraph of $G_{k+1}$, and there exists an integer $n>k$ such that every red-blue coloring of $G_n$ contains a monochromatic copy of $G_k$. Among the wide range of open problems in Ramsey theory, an interesting open question is ``Does there exist an ascending sequence $\{G_k\}$ with ${\lim }_{k\to \mathrm{\infty }}\chi (G_k)=\mathrm{\infty }$ and ${\lim }_{k\to \mathrm{\infty }}\omega (G_k)\mathrm{\ne }\mathrm{\infty }$ that is a Ramsey sequence?". In this paper, we solve this problem by constructing a Ramsey sequence $\{G_k\}$ with a bounded clique number such that ${\lim }_{k\to \mathrm{\infty }}\chi (G_k)=\mathrm{\infty }$. Furthermore, using the observation that any monotonic increasing sequence of graphs that contains a Ramsey sequence as a subgraph is also Ramsey, we can generate infinitely many Ramsey sequences using this example.