AI 中文总结
该研究证明了s≥3时Zarankiewicz数的Roman界在设计阈值正下方大部分范围无法达到，明确了同余障碍条件，并关联线性规划分析了相关界的可达性。
AI 中文摘要
设z(m,n;s,t)为m×n的0-1矩阵中不含s×t全1子矩阵的最大1的个数。Roman在1975年提出的不等式仍是s≥3时最优的通用上界，但在设计阈值T=(t-1)C(m,s)/(s+1)正下方的大部分范围内无法达到。证明分为两步：首先，对于n=T-c（1≤c≤sT/(s+2)），Roman界等于初等计数界(s+1)(T-c)+floor(2c/s)，这一步仅借助预算不等式和凸性即可得到，未引入额外内容；其次，达到该界要求除至多1列外，其余所有列的大小均为s+1或s+2，且每列中存在一个点，其所属s元组的数量可被d=gcd(s,C(s+1,2))整除，这使得剩余覆盖量被限制在模d的单一剩余类μ中，而全局计数可排除该情况。结合r=c mod s和仅依赖s的松弛量σ(r)，我们证明当1≤σ(r)<d且m·μ≠s·σ(r)，或μ≠0且m·μ>s·σ(r)时，z(m,T-c;s,t)≤Rom(m,T-c)-1。第一种情况是局限于一个剩余类的奇数现象，对应n取m^s量级的值；第二种情况要求μ≠0，但当m超过仅依赖s和μ的阈值时，可覆盖整个区间。对于s=4、t=2、m=28的情况，该结论覆盖全部2730个n值；当c=(s+1)/2且存在s-(m,s+1,t-1)设计时，z=(s+1)(T-c)恰好成立。最后，我们将该障碍与线性规划关联：所有2^m子集变量的松弛在对称化下会坍缩为计数界，因此仅覆盖约束的线性松弛无法超越Chen、Horsley和Mammoliti（arXiv:2310.12685，题为《三系阈值附近的Zarankiewicz数》）的界；对于Davies、Gill和Horsley的改进规划，其最优解仍在显式子族上达到Roman顶点，我们记录了该规划表现更优的情况。
英文摘要
Let z(m,n;s,t) be the largest number of ones in an m x n zero-one matrix with no s x t all-ones submatrix. Roman's 1975 inequality remains the best general upper bound for s>=3, but it is not attained on a large part of the range just below the design threshold T=(t-1)C(m,s)/(s+1). The proof has two steps. First, for n=T-c with 1<=c<=sT/(s+2), Roman's bound equals the elementary counting bound (s+1)(T-c)+floor(2c/s), adding nothing beyond a budget inequality and convexity. Second, attainment forces all but at most one column to have size s+1 or s+2; each such column has a point lying in a number of s-sets divisible by d=gcd(s,C(s+1,2)), pinning the leftover coverage there to a single residue mu mod d, which a global count rules out. With r=c mod s and slack sigma(r) depending only on s, we prove z(m,T-c;s,t) <= Rom(m,T-c)-1 whenever 1<=sigma(r)<d and m*mu != s*sigma(r), or mu != 0 and m*mu > s*sigma(r). The first case is an odd-s phenomenon confined to one residue class, giving order-m^s values of n; the second needs mu != 0 but covers the whole interval once m exceeds a threshold depending only on s and mu. For s=4, t=2, m=28 it covers all 2730 values of n; when c=(s+1)/2 and an s-(m,s+1,t-1) design exists, z=(s+1)(T-c) exactly. Finally we relate the obstruction to linear programming: the relaxation over all 2^m subset variables collapses, under symmetrisation, to the counting bound, so no linear relaxation of the covering constraints alone can beat the bound of Chen, Horsley, and Mammoliti (arXiv:2310.12685, "Zarankiewicz numbers near the triple system threshold"). For the refined program of Davies, Gill, and Horsley, its optimum is still attained at the Roman vertex on an explicit sub-family, and we record where their program does better.
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