---
tag: 12HP7ZS4D5
kind: conjecture
name: Sumner's conjecture
sources:
  - wikipedia: en:28866227@r1315922689
    locator: Sumner's conjecture
    note: adopted
    mention: wikipedia://en/28866227
    sections: [Statement]
same_as:
  wikidata: Q7637935
  wikipedia: Sumner's conjecture
inventory: [TZJPH47CRZ]
---

## Statement

**Sumner's conjecture** (also called **Sumner's universal tournament conjecture**) is a conjecture in extremal graph theory on oriented [[trees|062D9KXJB1]] in tournaments. It states that every orientation of every $n$-vertex [[tree|062D9KXJB1]] is a subgraph of every $(2n-2)$-vertex tournament. David Sumner, a graph theorist at the University of South Carolina, conjectured in 1971 that tournaments are universal graphs for polytrees. The conjecture was proven for all large $n$ by Daniela Kühn, Richard Mycroft, and Deryk Osthus.