Stability of Erdős–Ko–Rado theorems in circle geometries

Adriaensen, Sam

doi:10.1002/jcd.21854

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2023

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“…While finalizing our manuscript, a stronger stability version of the above mentioned result for k = 2 was published by Adriaensen; see [2]. Our proof is different, based on polynomials and hence might be of independent interest.…”

Section: Accepted Manuscriptmentioning

confidence: 88%

Intersecting families of graphs of functions over a finite field

Csajbók

Aguglia²,

Weiner³

2023

Ars Math. Contemp.

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Let U be a set of polynomials of degree at most k over F q , the finite field of q elements. Assume that U is an intersecting family, that is, the graphs of any two of the polynomials in U share a common point. Adriaensen proved that the size of U is at most q k with equality if and only if U is the set of all polynomials of degree at most k passing through a common point. In this manuscript, using a different, polynomial approach, we prove a stability version of this result, that is, the same conclusion holds if |U | > q k − q k−1 . We prove a stronger result when k = 2.For our purposes, we also prove the following results. If the set of directions determined by the graph of f is contained in an additive subgroup of F q , then the graph of f is a line. If the set of directions determined by at least q − √ q/2 affine points is contained in the set of squares/non-squares plus the common point of either the vertical or the horizontal lines, then up to an affinity the point set is contained in the graph of some polynomial of the form αx p k .

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Section: Accepted Manuscriptmentioning

confidence: 88%