Sumner's conjecture for oriented trees

Let TT be an oriented tree, let n=V(T)n=|V(T)| be its order, and let unvd(T)\operatorname{unvd}(T) denote its unavoidability.

Sumner's conjecture. Every oriented tree of order n>1n>1 is (2n2)(2n-2)-unavoidable; equivalently,

unvd(T)2n2.\operatorname{unvd}(T)\leqslant 2n-2.

This is a celebrated conjecture about tournaments and oriented trees. The source does not state a resolution, so its status remains open.

Sources & referencesView supporting material

Primary source

Pierre Aboulker, Frédéric Havet, William Lochet, Raul Lopes, Lucas Picasarri-Arrieta and Clément Rambaud, “Blow-ups and extensions of trees in tournaments”, arXiv:2410.23566 (2024).

Additional references

5 papers in this index state this conjecture (2013–2024). The statement above is taken from the most recent of them; the others are arXiv:2310.18719, arXiv:1912.04004, arXiv:1812.05167, arXiv:1307.4803.

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