Ed Miliband called today for a cap on media ownership and suggested it should be set lower than the proportion of the market currently owned by the Murdoch empire.
The Labour leader told the Leveson Inquiry he had "no worries" about a company owning 20% of the British market but said there was a "question of between 20-30%".
News International held 37% of the market until the closure of the News of the World but still retains a 34% share, the inquiry was told.
Mr Miliband told Lord Justice Leveson he did not believe one person should control 34% of the British press.
He said his "strong instinct" was that it was "too much" he wanted to see the inquiry look at a lower limit.
He added: "There's a question about what the limits should be. I should say we have no worries about someone owning 20% of the news market. I think there is then a question of between 20-30% where you should set a limit.
"That is where I'm coming from."
He told the inquiry his aim was not to "stifle one news organisation".
"My aim is plurality and a sense that... one organisation does not exert an overweening power."
Mr Miliband also said he believed some form of statutory regulation of the press was needed. But he did not believe that should be used to tackle newspaper content.
He said: "I think what we need on redress is something which is independent of the press and politicians, something which is comprehensive covering all newspapers, magazines and there is a clear question about internet organisations, something which is accessible, provides fastrack justice or redress for individuals."
He added: "I haven't yet seen a way forward without some kind of statutory support for the system.
"I think it would be very important to insert in any Bill constitutional safeguards on the freedom of the press."
Mr Miliband insisted he "would not countenance" any measures being used as a "licence for some massive bureaucratic assault on the press".
Lord Justice Leveson mooted the idea of limiting his "aspirations" in the area of media ownership to setting out the concerns that various witnesses have expressed and the counterbalancing arguments, and suggesting "appropriate authorities examine the position".
He said: "I'm not trying to shirk my responsibility but nor am I trying to bite off more than I could or should legitimately take on."
Mr Miliband said his own suggestion was that if a politician wanted to depart from recommendations made by the Competition Commission or broadcast regulator Ofcom, their decision could be challengeable.