A number of actions have been launched attempting to prevent the government from formally triggering Brexit without Parliament's authorisation.
During the opening hearing, government lawyers told the High Court Prime Minister Theresa May did not intend to trigger Article 50 of the Lisbon Treaty before the end of 2016.
Article 50 begins the Brexit process.
A hearing in October will allow time for a possible appeal to be completed before the government seeks to leave the EU in line with the 23 June referendum result, the court heard.
The hearing will take place over two to three days.
Government lawyers are expected to argue that the prime minister can use historic Royal Prerogative powers to start the process of withdrawing from the EU, a course the challengers say is unlawful.
They say Parliament must give its authorisation.
The judges heard that one of the law firms involved, Mishcon de Reya, had received letters of abuse which led to potential clients who had wanted to join the action withdrawing their names.
"It is racist abuse, it is anti-Semitic abuse and it is objectionable abuse," Lord Pannick QC, instructed by Mishcon, told the court.
The judges ruled that the lead case in the action should be that of Mishcon client Gina Miller, 51, an investment manager and philanthropist living in London who voted Remain in the EU referendum.
Other applicants include London hairdresser Deir Dos Santos, 37, as well as Britons living in France campaigning as Fair Deal for Expats.
Mrs May has said she will not trigger Article 50 - which sets in place a two-year process for leaving the EU - until the end of the year.
The recently-appointed Brexit Secretary, David Davis, has said Brexit should be triggered "before or by the start of next year".
EU leaders have urged the UK to do so as soon as possible.

Sokol Muja, 27, was backing a people carrier out of a bay at a car wash on Llantarnam Industrial Estate in Cwmbran when he hit Dave Fields, 75.
He admitted causing death by careless driving at Cardiff Crown Court.
Judge Eleri Rees gave Muja, from Cwmbran, a six-month suspended sentence and banned him from driving for two years on Friday.
Mr Fields had taken his car to be cleaned and was stood waiting when Muja reversed into him, causing him to fall and hit his head. He died in hospital five days later.
Muja also admitted driving without insurance and in accordance with the provisional licence he holds.
He was ordered to carry out 250 hours of community work and told to pay Â£350 in costs and a victim surcharge.
Judge Rees said: "Your action resulted in a tragic and needless loss of life and you will have to bear the burden of guilt throughout your life."
Afterwards, Mr Fields's daughter-in-law Kathy said: "He deserved to go to prison for what he did but I suppose we just have to accept it.
"It seems he paid just a small price for the immense harm he has caused."
A legal challenge over the UK leaving the EU will be heard by the High Court in October, two judges have decided.
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A learner driver who knocked over and killed a pensioner at a car wash has been given a suspended jail sentence.