2024/09/26 by Salia, Nika, Tóth, Dávid · 1 citation
#11T06 05D05 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2409.17821
This paper establishes an analog of the Erdős-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A k-uniform family of subsets of a set of finite size n is l-intersecting if any two subsets in the family intersect in at least l elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős-Ko-Rado theorem, which establishes a sharp upper bound on the size of these families. As an analog of the Erdős-Ko-Rado theorem, we determine the largest possible size of a family of monic polynomials, each of degree n, over a finite field Fq, where every pair of polynomials in the family shares a common factor of degree at least l. We establish that the upper bound for this size is qn-l and characterize all extremal families that achieve this maximum size. Further extending our study to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least l, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most n, we affirm that only trivial families achieve the corresponding upper bound.