For every integer n≥ 1, determine the exact value of operatornameex_2^+(n,RT_5)=max∑_v∈ V(D)d_D^+(v)^2: |V(D)|=n and D contains no copy of RT_5, where RT_5 is the regular tournament on five vertices. Iľkovič conjectured an explicit formula for this extremal quantity.

Iľkovič’s degree-square Turán conjecture for RT₅

For every integer $n\ge 1$ , determine the exact value of $\operatorname{ex}_2^+(n,\mathrm{RT}_5)=\max\left\{\sum_{v\in V(D)}d_D^+(v)^2:\ |V(D)|=n\text{ and }D\text{ contains no copy of }\mathrm{RT}_5\right\}$ , where $\mathrm{RT}_5$ is the regular tournament on five vertices. Iľkovič conjectured an explicit formula for this extremal quantity.

References

Primary source

arXiv

Additional references

Degree-square Turán problem for two self-converse tournaments — arXiv — Zhuoran Han, Yaojun Chen

directed-graph-theory extremal-graph-theory

Posted 6 days ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new preprint claims to settle the conjecture for the five-vertex regular tournament, while the wider classification remains open.

Iľkovič’s conjecture predicts the exact extremal degree-square quantity for digraphs avoiding the regular tournament $RT_5$ . An earlier Iľkovič preprint gave exact values and a conjecture for $RT_5$ , rather than a proof.

October 1, 2026 preprint

Zhuoran Han and Yaojun Chen claim to determine the extremal quantity for $RT_5$ , including the conjectured formula, as well as for transitive tournaments. This would settle the stated conjecture, but the retrieved evidence is a preprint announcement without independent mathematical verification; the broader classification of self-converse tournaments remains open.

Current status (as of October 2026): The $RT_5$ conjecture is claimed solved by Han and Chen’s October 1 preprint, but remains unverified; the broader self-converse-tournament classification is open.

Sources

No solutions have been posted yet.