Erdős Problem #74: locally almost bipartite graphs of infinite chromatic number

Astra proves that there exists a divergent function $$f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty,$$ such that no graph $G$ of infinite chromatic number can satisfy $$d_{\mathrm{bip}}(H)\le f(n)$$ for every finite $n$-vertex subgraph $H\subseteq G$, where $d_{\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite. In fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.

Astra proves that there exists a divergent function $$f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty,$$ such that no graph $G$ of infinite chromatic number can satisfy $$d_{\mathrm{bip}}(H)\le f(n)$$ for every finite $n$-vertex subgraph $H\subseteq G$, where $d_{\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite. In fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.