On a generalization of Ramsey numbers

Given the integers 1,, k,,1 2 , k 2 , r, which satisfy the condition 1,, 1 2 > r> k,, k 2 > 0, we define m = N(1,, k, ; 1 2 , k 2 ; r) as the smallest integer with the following property : if S is a set containing m points and the r-subsets of S are partitioned arbitrarily into two classes, then for i = 1 or 2 there exists an l i subset of S each of whose ki-subsets lies in some r-subset of the ith class . The integers defined in this way form a collection of which the usual Ramsey numbers are a special case : i .e ., the Ramsey number N(1 1 , 1 2 ; r) is represented as N(l,, r; 1,, r; r) . We derive two major results concerning the values of these generalized Ramsey numbers . If k, + k 2 = r + 1 then N(1,, k, ; 1 2 , k 2 ; r) = l, +12-k,-k2+1, corresponding to the "pigeonhole principle" . For k,+k i < r, we show that N(1 1 , k, ; 1 2 , k2 ; r) = max (1„ l 2 ) . The next interesting case occurs for k, + k 2 = r + 2, where we show that there are constants c, and c 2 such that for sufficiently large 1, 2 c,1< N(l, k, ; 1, k2 ; r) < 2c 2 1 . Given integers li , ki , i = 1, . . ., n, and r, which satisfy the properties l i > r > ki > 0, for i = 1, . . ., n, we may define an integer N(l 1 , k 1 ; 12 , k2 ; . . .,"n, kn ; r) = in as the smallest integer with the following property : If S is a set containing in points and the r-subsets of S are partitioned arbitrarily into n classes, then for some i, 1 < i < n, there exists an l i subset of S each of whose k i -subsets lies in some r-subset of the ith class . The fact that such an integer exists follows immediately from the existence of the Ramsey number N(l 1 , 1 2 , . . ., I n ; r), for if the set S contains this many points, there is some i, 1 < i < n, such that all the r-subsets of some li -set are of the ith class [3] . Then certainly each ki subset of this 1i subset lies in such an r-set, since k i < r . In what follows we shall be con* This work was supported by the Office of Naval Research Grants N00014-67-A-0204-0034 and N00014-70-A-0362-0002 . 30 P. Erdős, P.E. O Neil, On a generalization of RamseV numbers cerned mainly with the case where there are only two classes of r-subsets (n = 2) . The proof of the following remarks are entirely analogous to those found in [ 31 and will be omitted . Remark 1 . N(r, k l ; 1, k2 ; r) =N(1, ki ; r, k 2 ; r) =1 . Remark 2 . N(l i , k ] ; 1 2 , k 2 ; r) < N(N(1 1 -1, k 1 ; 1 2 , k2 ; r), k 1 -1 ; N(l 1 ,k i ;1 2 -1,k2 ;r),k 2 -1 ;r-1)+1 . The following remark has no counterpart in any theorem on Ramsey numbers, but is elementary . Remark 3. If ki < ki and kz < k 2 , then N(l 1 , ki ; 1 2 , k 2 ; r) < N(l 1 , k i ; 1 2 , k 2 ; r) . Proof. Let m < N(l 1 , k i ; 1 2 , k 2 ; r) . Then there exists a partition of the r-subsets of the m-set S into two classes such that every li subset contains a k i subset all of whose containing r-subsets are class j, j ~ i, i, j = 1, 2 . Since k'l < k i , k 2 < k 2 , the above property is inherited and m < N(l i , ki ; 1 2 , kz, r) . Thus N(1 1 , k',,"2, k2 ; r) <N(1 1 , k l ; 1 2 , k 2 ; r) .