The party - which was the Conservatives' junior coalition partner from 2010 to 2015 - created one of the surprises in last week's snap election, increasing their number of MPs from nine to 12.
Video: Who is the DUP leader, Arlene Foster?
As Mrs Foster met with her MPs in Westminster, she said: "The future's bright", prompting Ian Paisley junior to respond: "The future's orange".
Until she does, she and her ministers are on borrowed time.
Jeremy Corbyn, leader of the opposition Labour Party, which saw its number of parliamentary seats and share of the vote increase, said there could be another election this year or early in 2018 after last Thursday's vote produced no clear victor.
That would give May enough votes to pass her legislative agenda through parliament and govern, albeit with a wafer-thin majority.
Major warned that although Northern Ireland was a long way from returning to the violence that killed 3,600 people, he believed the peace process remained fragile almost two decades after a US -brokered 1998 peace agreement.
British Prime Minister Theresa May attempts to form an alliance to cling on to power before heading to Paris on Tuesday, leaving the EU's dismayed Brexit negotiator Michel Barnier wondering when exit talks will begin.
Seven MPs were elected in Sinn Féin colours last week, although the party has always refused to take up seats in Westminster.
"This new arrangement is very unsettling and people are concerned and anxious about what it may mean", Sinn Fein MP Michelle Glidernew told AFP. "There's a lot of anxiety", Sinn Fein MP Michelle Glidernew told AFP.
US Federal Reserve Raises Interest Rates, Unveils Balance Sheet Cuts for 2017
KEEPING SCORE: Japan's Nikkei 225 stock index fell 0.2 percent to 19,838.86 and South Korea's Kospi sank 0.6 percent to 2,358.66. Household spending has picked up in recent months, and fixed investment by business has continued to expand.
"A fundamental part of that peace process is that the United Kingdom government needs to be impartial between all the competing interests in Northern Ireland", he told the BBC.
"She said: 'I'm the person who got us into this mess and I'm the one who is going to get us out of it, ' " said one Conservative lawmaker who attended Monday's meeting.
"My preoccupation is that time is passing - it's passing quicker than anyone believes".
Next Monday has also been scheduled for the launch of Brexit talks between the Government and the other 27 European Union nations.
Brexit minister David Davis plans to go to Brussels on Monday to start the talks, which will reshape not only Britain's role in the world, but also that of a bloc praised by many for ensuring peace after World War Two.
Sinn Fein has declined to say whether it might challenge a Tory-DUP tie-up in the courts, but Alex Maskey, a Sinn Fein member of the Belfast Assembly, said it could "prove to be reckless".
The decision to announce a new date for the Speech, two days later than its scheduled date of June 19, was made after Mrs May's regular audience with the Queen on Wednesday.
Then, when the Brexit deal is finally done, Mrs.
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