More than 450,000 extra primary pupils will need places in England by 2015 as schools face a surging birth rate, government figures reveal.
This will place intense pressure on schools, particularly in urban areas.
In Barking there is a forecast for the primary school population to increase by more than 40% - the equivalent of dozens of new schools.
But a spokesman for the Department for Education says there are no plans to remove infant class size limits.
The government has released its latest figures on school capacity - including local authority forecasts for how demand for places is set to change between the school years 2010-2011 and 2015-2016.
This shows a picture of soaring demand for primary places in some areas, within a projected national increase in the number of primary-age pupils of 454,800.
This reflects a sustained population boom - with the birth rate in 2010 20% higher than in 2002.
London already faces a squeeze on places, with temporary "bulge" classrooms being erected in some schools - but this is set to intensify.
Barking will need to accommodate more than 8,000 extra primary pupils; Brent and Newham more than 6,000.
County councils, often with more pupils than urban authorities, also face big increases - Lancashire is forecasting demand to rise by 13,000, Hampshire by 11,000 and Kent is expecting to need places for more than 9,000 extra pupils.
As well as the shortage of places in some areas, there are still large numbers of unfilled places elsewhere - either because of demographic changes or because parents are not sending their children to unpopular schools.
The latest figures show that nationally there are more than 444,000 empty primary places - but not necessarily in the places where there is the growing demand.
This will be a tough planning challenge for national and local government, against a background of spending constraints and growing numbers of schools moving outside of local education authority control.
The rapid rise in the birth rate in some areas is going to require a substantial increase in capacity - with implications for buildings, playgrounds and staffing.
For example, the north London authority of Brent faces a primary school population increase of more than 25% by 2015.
Building company Wates said that in terms of the time needed to open a new school, it required an urgent response if places were to be ready in the next three years.
"While that might seem like a long way off, it typically takes at least two years from the time a decision is made to build a school before it is ready to open as a school," said Steve Beechey, head of education at Wates.
"So there is an acute need for new school building projects to get under way now to avoid a potentially critical shortfall of places in densely populated areas within the next few years."
As schools face this population surge, the school system is under tough financial constraints.
Capital spending was cut by 60% for the period of the comprehensive spending review.
But the government says that it is targeting extra funding at areas that are most under pressure.
Schools Minister Lord Hill said: "We're creating thousands more places to deal with the impact of soaring birth rates on primary schools.
"We're more than doubling targeted investment at areas facing the greatest pressure on numbers - to over £4bn in the next four years.
"We are building free schools and letting the most popular schools expand to meet demand from parents."
Breaking down this £4bn figure, the Department for Education says this represents £800m per year plus £500m extra for 2012-13 and a further £600m for 2013-15, announced in the Chancellor's autumn statement.
In terms of the demands on this money, the department suggests that a 400-pupil primary will cost in the region of £4m to £5m.
Steve Reed, executive member for children and young people at London Councils, said he welcomed the recognition that schools in the capital faced a "dramatic surge in numbers".
But he said: "The government announced before Christmas that London would receive extra funding, but this will still be less than we actually need."
Local authorities have a statutory duty to provide places for children - but they are also facing major structural changes in planning local education services.
Academies are their own admission authorities - operating outside of the local authority system - and it is expected that in the course of this year a majority of secondary schools will have become academies.
More primary schools are also set to become academies - adding another dimension to strategic planning for places.
The Local Government Association, responding to the new figures, called for better forecasting methods for future demand, to identify where the big increases are likely over five and 10 years.
This rise in demand is also likely to raise questions about the maximum class size limits for infant classes.
This was a flagship policy of the Labour government - setting a legal upper limit of 30 pupils for infant classes in primary school.
Class size limits
The Department for Education commissioned a study of the evidence of the importance of class sizes - with the results published on 22 December.
The findings echoed international reports which have questioned the link between class size and standards.
It concluded that "class size reduction policies are not the best option in terms of value for money to raising pupil attainment, compared to others such as increasing teacher effectiveness".
"Broadly evidence suggests that class size reduction policies have an uncertain and diminishing effect on pupil achievement in the long run."
But it noted that a "smaller class size has a positive impact on attainment and behaviour in the early years of school".
And it also acknowledged the strong parental support for smaller classes.
In terms of the international context, the report says that schools in England already have unusually big primary class sizes for a developed country - but suggested that there was no clear correlation between average class size and attainment.
But a Department for Education spokesman emphasised that there were no plans to remove the maximum class size.
"No parent wants their child taught in huge classes - so it's right local authorities create extra places to keep sizes down and relief pressure on places," said a department spokesman.
"It remains illegal for infant classes to exceed 30 pupils per teacher. We take a tough line with any breach and pretty much every class remain below the statutory limit - a level that has remained steady for the last few years."