Parallelism in a restricted linear space

Discrete Mo ehematics 0 North-Holland Pub1

6) 395-398. omyany

In this papertwo tines are called parallel if they are identical or if they have no point in cornm~rr, It is then shown that in a restricted iineu space of sqwrc order n paraklism is an equivaisnce relation on the n-lines.

In this paper we will use the terminology and notation of [ 3 1. We wiil also refer to the prerequisites Pl to Pa and Theorems Tl to T3 listed there. Since this paper is a continuation of the work, of /3 ] and [4] bthe lemmas and theorems will be named accordingly. In the development of certain branches af linear spaices an important notion has been that of parallelism. Two lines are said to be parallel if they are identical or have no peint in common. The case when parallelism is an equivalence relation on the entire set of lines has already been studied and indeed been completely so!ved (see f 1,2] ). In this paper we will display for any MS except one a set of lines such that on this set of lines parallelism is an equivalence relation.

implies

that

ail points on x, are of degree at least n f 2. So P4 yields

p f tl -I2y--

1 3 s,

+ (n

+ 1 pi

a

(n

f-

l)?? ,

r12-k 1. Moreover, since the corollary t f t2 + l,wehaves, < rs.Onthe other hand ts nut on x, arcsof degree at least n + 1. Hence we hwe c(r-2+ 1) + (y - C)??a

c

(9,

3

a

;

’

C

h,

3

ntf2 + 2)+(g

-

n)(rz +

lj 3

a n+lbyPH.

we see that the w jression Z,(b, - CI, - l)r, nes 1 &ch meet x, (exluding x, itself) and miss he meet Lx-, a d x,. Then there are at least 4, (a, - 1) o to x, that do not ass through u. At most (a, - I)& -.-1) ese lines also meet A-, since none pass through u. Hence at least flp + P ) of the lines that meet x, anQ x, miss .xp, and we

pertiesare t~vial~y

true. LS of’ square order n o Lin’s WOSS,and suppose that r-araklism were not transitive on its set of Mines. So there would be n-lines x,, xp and x would meet in u, say, but would both miss x,. shows at once that A? could not be an FAP, an FSP3 or a pun.ctured FSP3, Neither couM it be a near-pencil. So Theorems C and D and the corollary to Theorem C yield aI = IZ+ 1 b, for all U, and p < 112 + e + I. over, the corollary to Lemma 6 above yields

By Lemma 5 there is an (I? + 1 )-line that meets x,. Let y be such an (pz+ 1 )-line. Now through every point ofy not lying on x’, there is at least one line that misses x, (by P3), whence s, 2 II. Now by applying PS to x, we obtain

;;tpY2ah

- 1.

Thus p 3 n2 + I?.

Ckw (i): X, corz~a~~rts iltz Irr + 1)-point. . applying P2 to up yields

=n2 -0

-

Let

u,

be such a point. Then

1.

Thus p = .* + n and the above inequality becomes an equality, which implies there are IZ(rz f 1)-lines together with s, passing throu Therefore, the line joining tip to ts is an (rt + 1).line, say x,. Q

CbQ

I_

there are at least II et

sinceb(u) 2 =P

.

n +- 2.

1)P,, 3 n -- 1 ,

n1

lines th;at meet x, a

iss s, .

implks p + ~2- 1 3 y - It 2~ n* + 2it. There+r~t+ 1. Sinceq = n ,; 1) we an conclude from aint lies only on (n + t )-lines. Tkxefure, pvery point of xc; has assing t~u~~h evey point of x, ahere is at ee at least fl + 2. tslelf that misses x,. Tkrefore s, ra + 1. st one tine other t bs for the line x, we have from P4 thrltt n+l+d+

C(h,--n--

xp,,

a

1, nee ,P 2~n z + n + 2, which

final con&3

Paul de Wittt: for pointing

ffine Ebenen, Arch Math. (‘Barni)13 (I 962) 1 itte, On 8 Paschian condiciczzfor tineslur spaces,

sim-