For every integer k≥ 2, if A⊆ 2^[2k] is an intersecting temperate family, then | A|≤ binom2k-1k-1+binom2kk+1+binom2k-1k+2. Moreover, equality is conjectured to hold only for a family of the following form: there exists an element a∈[2k] such that A=A∈binom[2k]k:a∈ Acup binom[2k]k+1cupA∈binom[2k]k+2:a notin A.

Petr–Turek conjecture on intersecting temperate families

Petr and Turek conjectured the extremal size and structure of intersecting temperate families in the even case. The conjectured extremal family is a lightning, consisting of three consecutive set layers.

Known results

For odd $n=2k-1$ , Petr and Turek proved the maximum and, for $n\ge 5$ , unique extremal family in 2024.
For even $n=2k$ , their 2024 paper stated the maximum and lightning structure as Conjecture 1.7 for general families.
Before 2026, the even-case result was only proposed under restriction to the three middle layers.

2026 three-layer preprint

Kada Williams's 2026 preprint reports a proof for families contained in $\binom{[2k]}{k}\cup\binom{[2k]}{k+1}\cup\binom{[2k]}{k+2}$ . It claims the conjectured maximum and equality classification: all $k$ -sets containing a fixed element, all $(k+1)$ -sets, and all $(k+2)$ -sets avoiding it. The broader even-case conjecture is explicitly only partially resolved, and the claim has not been independently verified.

Current status (as of September 2026): The three-layer even-case classification is claimed proved, but the broader even-case conjecture remains open and the preprint's proof is unverified.

Petr–Turek conjecture on intersecting temperate families

Primary source

Additional references

Progress summary

Solutions 0