Traces Without Maximal Chains
Ta Sheng Tan

TL;DR
This paper proves Patkós' conjecture that for large n, the maximum size of a family of k-sets with traces on r-subsets avoiding maximal chains is bounded by a specific binomial coefficient.
Contribution
The paper confirms Patkós' conjecture, establishing an exact upper bound on the size of such set families for large n.
Findings
Proves Patkós' conjecture for large n.
Establishes the maximum size of set families avoiding maximal chains in traces.
Provides combinatorial bounds related to set family traces.
Abstract
The trace of a family of sets on a set is . If is a family of -sets from an -set such that for any -subset the trace does not contain a maximal chain, then how large can be? Patk\'os conjectured that, for sufficiently large, the size of is at most . Our aim in this paper is to prove this conjecture.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory
