Soon after former Fairfax County School Board member Janet S. Oleszek clinched victory in the Democratic Party primary to run for the Board of Supervisors’ Braddock District seat, Republican incumbent John C. Cook offered an open hand of welcome, more or less. The GOP county chairman delivered a few slaps.

Cook, in a written statement issued Wednesday, said he called Oleszek to congratulate her on her victory and also proposed a series of neighborhood debates around the district. What Cook has in mind is 10 forums — one a week — and he said he could begin as soon as this weekend. His statement also contained a note or two of criticism, wondering whether Oleszek has performed any public service in the community since leading the PTA 10 years ago before her service on the school board. He also challenged Oleszek for specifics on fixing transportation problems.

About the same time, Anthony Bedell, chairman of the Fairfax County Republican Committee, put out a more sharply worded statement accusing Oleszek of running a campaign that lacks substance.

“Platitudes like ‘improving transportation’ or ‘supporting schools’ offer no substantive ideas,” Bedell said.  “Everyone wants good schools and better transportation.  The question is, who has the ideas and the leadership skills to make these things happen?  Ms. Oleszek has offered no specifics. Not one.”

Her campaign quickly fired back.

“If John Cook wants to attack Janet’s 39-year record of public service in Fairfax County, he should man up and do it himself and not hide behind his local party chairman,” Benjamin Tribbett, Oleszek’s campaign chair, said.

Tribbett, who also writes the notlarrysabato blog, also confirmed Thursday that the candidates have agreed to 10 debates and were working out the arrangements for them. “Janet is thrilled to participate in all of them,” Tribbett said.

The two major party candidates in Braddock will also have some company on the campaign trail from Carey C. Campbell, a Springfield resident who has filed to run as an independent. Campbell, 54, an Air Force veteran, accountant and former vice president of the Fairfax County Federation Civic Associations, said also he wanted to debate Oleszek and Cook.

With the filing deadline now passed, Gary Scott, deputy registrar in the Fairfax County Office of Elections, said Thursday that other independent candidates for the Board of Supervisors include:

• Christopher F. DeCarlo, 52, who lives in the Mantua area, independent candidate for chairman at-large.

• A. Willard “Will” Radle, 39, of the Alexandria section of Fairfax County, independent candidate for chairman at-large.

• Glenda G. Parker, 64, of the Alexandria section of Fairfax County, Mount Vernon District supervisor. (Parker initially filed to run as chairman at-large but withdrew to run for the Mount Vernon seat, Scott said.)