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Statement

Setting (pp. 26--27). For $n,d\in\mathbb N$ with $2\le d\le n$, $H_{n,d}$ is the complete $d$-homogeneous hypergraph on an $n$-element vertex set $V$: its edges are all $d$-element subsets of $V$. Each edge $e$ carries a weight $w(e)\in\mathbb R$, and

$$\operatorname{disc}(H_{n,d}(W))=\min_\theta\max_{V'\subset V}\Bigl|\sum_{e\in E,\ e\subset V'}\theta(e)w(e)\Bigr|,$$

the minimum being over all colorings $\theta:E\to\{-1,1\}$.

Theorem 8 (p. 27). Let $n,d\in\mathbb N$, $2\le d\le n$. There are constants $c'_d$ and $C'_d$, independent of $n$ and of the weights $W$, such that

$$c'_d\sum_{v\in V}\Bigl(\sum_{e\in E:\,v\in e}w(e)^2\Bigr)^{1/2}\le\operatorname{disc}(H_{n,d}(W))\le\mathsf E_\theta\max_{V'\subset V}\Bigl|\sum_{e\in E,\ e\subset V'}\theta(e)w(e)\Bigr|\le C'_d\sum_{v\in V}\Bigl(\sum_{e\in E:\,v\in e}w(e)^2\Bigr)^{1/2}.$$

The middle inequality is immediate, a minimum being at most an average; the content is the two outer bounds.

Unit weights (p. 27). For $w\equiv1$ each vertex lies in $\binom{n-1}{d-1}$ edges, so the right-hand sum is $n\binom{n-1}{d-1}^{1/2}$, which the paper notes is of order $n^{(d+1)/2}$ with constants depending only on $d$. The theorem therefore contains the Erdős--Spencer estimates $c_dn^{(d+1)/2}\le\operatorname{disc}(H_{n,d})\le C_dn^{(d+1)/2}$, the paper's (1) (p. 1).

Other hypergraphs (p. 27). The paper remarks that the result extends immediately to arbitrary, not necessarily complete, homogeneous edge-weighted hypergraphs, as in the passage from Theorem 6 to Theorem 7 (zero weights on the missing edges). The introduction states the estimate in that generality as (2) (p. 2), there "for every $d\in\mathbb N$" (quoted); Theorem 8 itself assumes $2\le d\le n$.

Source. Sergey V. Astashkin and Konstantin V. Lykov, Random unconditional convergence of Rademacher chaos in $L_\infty$ and sharp estimates for discrepancy of weighted graphs and hypergraphs, arXiv:2412.20107v1 [math.PR], 28 December 2024; Section 6 (pp. 24--27), part (c), the definitions on pp. 26--27 and Theorem 8 on p. 27. The edition read is identified on the source card.

Read depth. Claims checked: the definitions, the statement and the unit-weight remark were read clause by clause on the page images. Nothing here is independently reviewed.

Proof pointer

P. 27. The discrepancy of a coloring is the multidimensional modified cut-norm (22) (p. 12) of the array $(\theta(e)w(e))$ indexed by increasing $d$-tuples, so the theorem follows from Corollary 8 (p. 24), the order-$d$ chaos form of Theorem 4. Corollary 8 bounds by the largest of the $d$ one-coordinate sums, which is within a factor $d$ of the vertex sum here (a check of this page). The paper obtains Corollary 8 from Theorem 4 and decoupling "precisely in the same way" (p. 24, quoted) as in the second-order case, without a separate written proof.

Dependencies

Corollary 8 (p. 24), from Theorem 4 (p. 19), the decoupling Corollary 1 (p. 8) and the equivalence (23) (p. 12).

Bears on

• Problem 1028: at $d=2$ with unit weights the theorem gives the unordered-edge discrepancy of $K_n$ of order $n^{3/2}$ for every $n\ge2$, the same consequence as Theorem 7, with constants that are not made explicit; it gives no leading constant or exact value.