Uniform Set Systems with Uniform Witnesses

) by Yang and Yu [12], matching the second order term of the family constructed in [10] up to a multiplicative constant. For more works along this line, see [4,9,11].

Notice that in the case when F is a (d + 1)-uniform family, we can equivalently formulate the condition that VC(F) ⩽ d as follows: for each F ∈ F, there exists a subset B F ⊆ F such that for any F ′ ∈ F, we have F ∩ F ′ ̸ = B F . We call B F a witness of F in the sense that it cannot be realized as an intersection of F with any set F ′ ∈ F. In this note, we are interested in the case where all witness sets B F have the same size. Definition 1.1 (s-witness family). Let n ⩾ d + 1 and 0 ⩽ s ⩽ d. A family F ⊆ [n] d+1 is an s-witness family if it satisfies the following condition: for every F ∈ F, there exists B F ⊆ F of size s such that F ∩ F ′ ̸ = B F for every F ′ ∈ F.

A 0-witness family F is exactly an intersecting family since we have F ∩ F ′ ̸ = ∅ for all F, F ′ ∈ F. The well-known Erdős-Ko-Rado Theorem states the following upper bound for intersecting families.

Theorem 1.2 (Erdős-Ko-Rado [6]). Let d ⩾ 0 and n ⩾ 2(d + 1). If F ⊆ [n] d+1 is an intersecting family, then |F| ⩽ n-1 d . Moreover, when n > 2(d + 1), equality holds if and only if F is a star. As a generalization of the Erdős-Ko-Rado Theorem, the first author, Xu, Yip, and Zhang made the following conjecture in [4].

). Let n ⩾ 2(d + 1) and 0 ⩽ s ⩽ d. Suppose F ⊆ [n] d+1 is an s-witness family. Then |F| ⩽ n-1 d . It is clear that when s = 0, Conjecture 1.3 follows from Theorem 1.2. In [4], the authors confirmed Conjecture 1.3 in the case where s ∈ {1, d} and n is sufficiently large. Moreover, they showed that equality holds if and only if F is a star when s ∈ {1, d} and remarked that it might be plausible to conjecture that the maximum set families achieving the bound in Conjecture 1.3 must be stars for all 0 ⩽ s ⩽ d.

In this note, we prove Conjecture 1.3 in the case when s ⩽ d/2 and n is sufficiently large.

Theorem 1.4. Let 1 ⩽ s ⩽ d/2 and let n be sufficiently large. Suppose F ⊆ [n] d+1 is an s-witness family.

, then F is a star. Together with Theorem 1.2 and the results in [4], the remaining open cases of Conjecture 1.3 are when d/2 < s ⩽ d -1. In these remaining open cases, we give a construction that is not a single star achieving the size n-1 d , illustrating the barriers to adapting current methods as all the proofs of the known cases show that F must be a single star if equality is achieved.

Theorem 1.5. For any n ⩾ s + 3 and d/2 < s ⩽ d -1, there exists an s-witness family F ⊆ [n] d+1 of size

that is not a star.

Since our proof of Theorem 1.4 proceeds by approximating F by a single star, it does not generalize easily to the cases where s > d/2. Thus, we believe new ideas are required to prove the remaining cases of Conjecture 1.3.

1.1. Paper Organization. In Section 2 we present the proof of Theorem 1.4. Then in Section 3 we give the construction that proves Theorem 1.5. Finally, in Section 4 we make some concluding remarks.

Notations. For sets A, B ⊆ [n], we denote their symmetric difference as A△B = (A \ B) ∪ (B \ A). We always consider s and d as fixed and n → ∞. We use O d (•) and Ω d (•) to suppress leading constants depending on d.

Acknowledgments. We thank the organizers of the SLMath workshop “Algebraic and Analytic Methods in Combinatorics” (March 2025) attended by the authors where this project was initiated. We would also like to thank Zixiang Xu for comments on an early draft of this note.

Our proof consists of three steps split into the three subsequent subsections. In the first step presented in Section 2.1, we analyze the structure of an s-witness set family F ⊆ [n] d+1 for 1 ⩽ s ⩽ d/2. In particular, we show in Lemma 2.2 that we can capture the intersection properties of F using set families of constant size. Then in Lemma 2.4, we define a certain injection of a subset of F into [n] d that will be the main object that we analyze later on. Next, in Section 2.2, we show that any family F t