Evasive sets, twisted varieties, and container-clique trees
Jeck Lim, Jiaxi Nie, Ji Zeng

TL;DR
This paper introduces new constructions of evasive sets in finite affine spaces and twisted varieties in projective spaces, providing bounds, enumeration results, and a novel container method with applications to combinatorial problems.
Contribution
It establishes the existence of large evasive sets with smaller parameters, introduces the concept of twisted varieties, and develops a new container method for combinatorial enumeration.
Findings
Existence of large evasive sets with size rom ^{n-k} o \u1d4f(q^{n-k})
Upper bound of 2^{O(q^{n-k})} on the number of evasive sets
A new container method technique applied to combinatorial enumeration
Abstract
In the affine space over the finite field of order , a point set is said to be -evasive if the intersection between and any variety, of dimension and degree at most , has cardinality less than . As tends to infinity, the size of a -evasive set in is at most by a simple averaging argument. We exhibit the existence of such evasive sets of sizes at least for much smaller values of than previously known constructions, and establish an enumerative upper bound for the total number of such evasive sets. The existence result is based on our study of twisted varieties. In the projective space over an algebraically closed field, a variety is said to be -twisted if the intersection between and any variety, of dimension $n -…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
