2025/11/18 by Balázs Patkós, Patkós, Balázs
Mathematics · #Advanced Topology and Set Theory #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #math.CO
paper · pdf · doi:10.48550/arxiv.2511.14298
openalex publication_date 2025/11/18 · openalex created_date 2025/11/20 · openalex updated_date 2026/07/28
A family G of sets is a copy of a poset (P,\leqslant) if (G,⊆) is isomorphic to (P,\leqslant). The forbidden subposet problem asks for determining La^*(n,P), the maximum size of a family F⊆ 2[n] that does not contain any copy of P. We study the rainbow version of this problem: what is the maximum size LaR^*(n,P) of a family F=∪i=1mAi such that all Ai are antichains and there is no copy of P with all sets coming from distinct Ai or equivalently F admits a proper coloring (sets F⊂ F' must receive different colors) with no rainbow copy of P. A poset (Q,\leqslant') rainbow forces (P,\leqslant) if any proper coloring c of Q (q\leqslant' q' or q'\leqslant' q implies c(q)≠ c(q')) admits a rainbow copy of P. We establish connection between the La^* and the La^*R functions via poset rainbow forcing, determine the asymptotics of LaR^*(n,T) for all tree posets and obtain further exact or asymptotic results for antichains and complete bipartite posets.