Open Problems around Uncountable Graphs

We collect some famous and some less well know open problems from the theory of uncountable graphs focusing mainly on chromatic number. This short note serves as a reference list for a talk given at the University of East Anglia, Independence Results in Mathematics and Challenges in Iterated Forcing workshop in November, 2015. We highlight that there are several very well written recent surveys on problems in infinite combinatorics and infinite graphs: Komjáth [18, 20], Todorcevic [32]. Although we mainly focus on uncountable graphs, there is significant work done on countably infinite graphs: see the survey of Halin [11] and the projects of the Hamburg school lead by R. Diestel [3]. We are grateful for the help of P. Komjáth, L. Soukup, S. Todorcevic and many others in preparing these notes. Basic definitions. A graph G is an ordered pair (V,E) such that E ⊆ [V ]. We let G denote the complement of G i.e. (V, [V ] \ E). We let NG(v) = {u ∈ V : {u, v} ∈ E} for v ∈ V . The chromatic number of a graph G is the minimal cardinal κ such that G admits a good colouring i.e. there is f : V → κ such that f(x) 6= f(y) whenever {x, y} ∈ E. The product G0×G1 of G0 = (V0, E0) and G1 = (V1, E1) has vertices V0×V1 and edges {(x, y), (x′, y′)} where {x, x′} ∈ E0 and {y, y′} ∈ E1. 1. Connected subgraphs A graph G is infinitely connected iff the removal of finitely many vertices leaves G connected. Conjecture 1.1 (Erdős, Hajnal). There is a graph G with chromatic number ω1 without non empty infinitely connected subgraphs. There is a graph with chromatic number ω1 without uncountable infinitely connected subgraphs [29] (but the known examples contain plenty of countably infinite complete subgraphs). A version of this problem first appeared in [6] and later in several surveys and problem sets [7, 18, 20]. Related results are proved in [15, 16, 19]. 2. Triangle free subgraphs Conjecture 2.1 (Erdős, Hajnal). There is a graph G with chromatic number ω1 such that every subgraph with uncountable chromatic number contains a triangle. Consistently yes (even K4 can be ommitted in G, while CH may or may not hold) [21]. It would be interesting to find more constructive examples, e.g. using ♦. The problem is still open in ZFC.