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Abstract Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $$\Delta $$ , then for every integer $$k \geq \Delta +1$$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $$1$$ ; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree $$\Delta $$ , then for each $$k \geq \Delta + 1$$ , G has a Borel proper k -colouring in which every two colour classes are related by an element of the Borel full semigroup of G . In particular, such colourings are equitable with respect to every G -invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $$\Delta $$ -colourings of graphs with small average degree. Namely, we prove that if $$\Delta \geq 3$$ , G does not contain a clique on $$\Delta + 1$$ vertices and $$\mu $$ is an atomless G -invariant probability measure such that the average degree of G with respect to $$\mu $$ is at most $$\Delta /5$$ , then G has a $$\mu $$ -equitable $$\Delta $$ -colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks’ theorem due to Kostochka and Nakprasit.
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Orienting Borel graphs
We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by k k for various cardinals k k . We show that for a probability measure preserving (p.m.p) graph G G , a measurable orientation can be found when k k is larger than the normalized cost of the restriction of G G to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every k k we also find graphs which admit measurable orientations with outdegree bounded by k k but no such Borel orientations. Finally, for special values of k k we bound the projective complexity of Borel k k -orientability for graphs and graphings of equivalence relations. It follows from these bounds that the set of equivalence relations admitting a Borel selector is Σ 2 1 \mathbf {\Sigma }_{2}^{1} in the codes, in stark contrast to the case of smooth relations.
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- Award ID(s):
- 1764174
- PAR ID:
- 10378353
- Date Published:
- Journal Name:
- Proceedings of the American Mathematical Society
- Volume:
- 150
- Issue:
- 754
- ISSN:
- 0002-9939
- Page Range / eLocation ID:
- 1779 to 1793
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/proc/15742
