Inferred children (per census): Charity Thompson (Hamby) and James Thompson. Children claimed by alleged descendants: Nathan Thompson, Mary Thompson (Reynolds), Sarah Thompson (Jones).
The theory that John Thompson who married Elizabeth Spencer is the father of William Thompson who married Elizabeth Lucinda Jones in 1839 in adjacent Caldwell County, Kentucky, is currently based on the following:
Multiple autosomal DNA matches between descendants at Ancestry, but no triangulation yet.
Proximity of Caldwell and Christian Counties.
Census numbers/ages fit (however, William can fit other Thompson households in the area)
Sarah, alleged daughter of John, married into the same Jones family that William did
Mary, alleged daughter of John, married into the same Reynolds family that William's wife's child by her second marriage did.
Sources
United States Federal Census. 1850. Kentucky, Christian County. Page 473b. Roll: M432_196; Image: 379. John Thompson household. Image of original document at Ancestry.com.
Kentucky, County Marriage Records, 1783-1965. Christian County. John Thompson and Elizabeth Spencer. 22 Jan 1800. Image of original document at Ancestry.com.
Kentucky, Death Records, 1852-1953. Christian County. John Thompson. 11 May 1853, age 78. Parents: Thomas and Mary Thompson. Image of original document at Ancestry.com.
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DNA Connections
It may be possible to confirm family relationships with John by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with John: