The ten, led by James Casey, are asking judge Mr Justice Burton to rule that the council's latest moves to evict them from Dalewood Gardens and Bewbush West Playing Fields are unlawful.
They claim among other things that the council's policy of evicting them from every site on which they encamp, subject only to urgent welfare needs such as imminent open-heart surgery, is unlawful.
The council deny this is their policy, but the travellers say that is belied by evidence dating back 16 years that shows they have been evicted from sites in the Crawley area up to 40 times a year.
The travellers say that, in reaching their latest decision to evict them, the council took into account welfare checks on them and their families that were 'riddled with inaccuracies and omissions'.
They claim that the council also failed to take into account a number of other factors, including:
The number of times the families had been evicted from sites in the borough, up to 40 times a year since they began coming to the area in 1992/1993; The amount of schooling the 25 school age children in the group have missed as a result of evictions, and will miss as a result of further evictions;
The cumulative impact on the health and welfare of the travellers, as a result of the problems of staying registered with doctors when forced to move in and out of catchment areas; The fact they would be left homeless with nowehere else to go; And the council's failure to provide any official sites in its borough despite the pressing need.
They also claim that the council failed to consider whether eviction would interfere with the travellers' rights to respect for their private and family life under Article 8 of the European Convention on Human Rights.
They argue that the council's repeated offers of only temporary bed and breakfast accommodation are inadequate, as a result of their cultural aversion to 'bricks-and-mortar' housing.
The hearing is expected to be adjourned after further argument today, and resume on 16 February.
The judge is then expected to reserve his decision in order to give it in writing. The travellers' current residence is protected by an interim court order pending the outcome of these proceedings.
The travellers named in court papers are: James Casey, Michael MacCarthy, Martin MacCarthy, Mary Maughan, Michael MacCarthy Jr, John Delaney, Patrick McCarthy, Clayton Maughan, Erica Ross and John Ross.
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