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The Rado path decomposition theorem. (English) Zbl 1429.05062
Summary: Let $$c : [\omega]^2 \rightarrow r$$. A path of color $$j$$ is a listing (possibly empty) of integers $$\{a_0, a_1, a_2, \dots\}$$ such that, for all $$i \geq 0$$, if $$a_{i+1}$$ exists then $$c(a_i, a_{i+1} = j$$. An empty list can be a path of any color. A singleton can be a path of any color. Paths might be finite or infinite. The color is determined for paths of more than one node. Improving on a result of P. Erdős (oral communication), R. Rado [Ann. Discrete Math. 3, 191–194 (1978; Zbl 0388.05031)] published a theorem which implies: Rado path decomposition: Let $$c : [ \omega ]^2 \rightarrow r$$. Then, for each $$j < r$$, there is a path of color $$j$$ such that these $$r$$ paths (as sets) partition $$\omega$$ (so they are pairwise disjoint sets and their union is everything). Here we will provide some results and proofs which allow us to analyze the effective content of this theorem.
##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C38 Paths and cycles 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
path of color
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