New Bounds for Limited Zarankiewicz Numbers from $K_{5t}$ Blocks

RESUMO

The restricted augmented Zarankiewicz number \(z_L(m,n)\) yields core combinatorial lower bounds for the maximal SOS rank of biquadratic forms. All previously known infinite admissible graph families rely on \(K_{4t}\) incidence graphs, attaining an asymptotic relative gap limit of \(1/4\). This work develops a new infinite family built from \(K_{5t}\) incidence bipartite graphs with \(\mathbb{Z}_5\) cyclic labeling for block partitions. We construct valid nondegenerate intra-block and inter-block 2-edges, derive a quadratic closed-form lower bound of \(z_L\), and prove its relative gap converges asymptotically to \(2/5\). Full enumeration for \(t=1\) verifies the exact value \(z_L(10,5)=23\). Under nondegenerate and generalized \(C_4\)-free constraints, the ratio \(2/5\) is shown to be the maximal asymptotic ratio attainable under this block framework. Our results expand the library of extremal bipartite graphs and sharpen lower bounds for biquadratic SOS rank, with further open problems for general \(K_{kt}\) constructions outlined in closing.