Znám, Štefan

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Generalisation of a Number-Theoretical Result

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Matematicko-fyzikálny časopisŠtefan ZnámGeneralisation of a Number-Theoretical ResultMatematicko-fyzikálny časopis, Vol. 16 (1966), No. 4, 357--361Persistent URL: http://dml.cz/dmlcz/126978Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1966Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czM A T E 3 I A T I C K 0 - F Y Z I K A L N Ý Č A S O P I S SAV, 16, 4, 1960 GENERALISATION OF A NUMBER-THEORETICAL RESULT ŠTEFAN ZNÁM, Bratislava Let k > 3 be a n a t u r a l n u m b e r a n d let M be a set of n a t u r a l n u m b e r s . We say t h a t M is a k-thin set if from t h e condi t ion a, , O2, . . . , (th i e M if follows t h a t Oi -\~ a2 + ... -\- (H-i $ M. W i t h other words: t h e set M is k-thin if in its n u m b e r s t h e equat ion (l[ -|- «•_> i . - . H OA- 1 ^ Oh is not solvable (the n u m b e r s O; can be equal) . Let us d e n o t e by J(k, p) t h e g rea tes t n a t u r a l n u m b e r for which there exist p disjoint k-thin sets Syi, S<>, .. ., ^ such t h a t |1,1>, . . . ,/(*,*>)} -- U*S';. T h e existence of J\k\ p) for a r b i t r a r y k a n d p follows from T h e o r e m s 3 and 4 of art icle [3]. T h e case k ---- 3 was t r e a t e d by I. S o h u r in art icle [4]. He proved n a m e l y t h e inequalit ies ( I ) / ( : , , / , l ) :> :,./(:,./») [ I , :{" l ( - ) / ( » . / > ) ; • . ; . , - • In our art icle we shall generalize t h e inequali t ies (1) a n d (2) for t h e case of an a r b i t r a r y k " 3 a n d show the i r appl icat ion t o t h e t h e o r y of g raphs . 35 7 Theorem 1. Lei k 3 find p be natund numbers. )Vc lair< (3) /(/%/>-'- I) Zk.f(k,p) I (k 2). N o t e I. For an a rb i t ra ry k 3 we have /(Z\ I.) I: 2 and therefore because of (3) f(k, p) V k.f(k, p I) -| (k 2) ::-. k* . / ( / • , p - 2) -,-- k(k 2) • (/• 2) . .; . . . - k? l . / ( / ' , I) !-- l> ^(k - 2) -f .. /'(/• 2) (/• 2) k- 2 (k 2)(/> i |- k/>^ ) . . . i k f 1) ,..- (/> I). k — 2 i. e. /(/", P) > ( ^ — 1) a 1 and th is is a general isat ion of t h e re la t ion (2) for a n y k : 3. N o t e 2. If in (3) we p u t k ----- 3, we get the re la t ion (I) . P r o o f of t he 'Theorem 1. From the definition oif(k\ p) it follows t h a t the re exist p disjoint k-thin sets S\, 82, . •., >SV such t h a t K,-*, • ••>/(*>/>)) --us,. i I Let us pu t V M - ! M P) -f- W W \ />) -1 * (/• - i ) / ( / - , />) ; (/• -T, From the inequal i ty (k \)\f(k,p) ! IJ .- ( / - \)fik\p) ; (/• 2) it follow* t h a t N,,: 1 is a k-thin set. Now, to accomplish tin4 proof", it is .sufficient to show t h a t the numbers (4) $k - $f(k,p) .- (k- \)\. $k I ) . / ' ( / • • f>) / | . •• I/'/(/',/>) !• (/•• -Ml (tin1 number of which is/(/•', p)) can be divided into <s'i, N-_>, .. . N., so tha t a.ftei add ing some numbers from (4) to Sf- we get again a. f-tliin .set .-1/ . Let us denote r/ (/•• $f(k,p) ! (k 2). i u v r y n u m b e r " from U) can he writ ten in the form O r(a) \ d, w h r t r c(/,'.) is a natural IIIIIUIMT fulfilling the condition o - . r(a) f(k, />). Now let us add each number a from (4) to t he same set to w Inch the numbei 3 5S c{a) belongs. The sets arisen in th i s way deno te by A\. J o , . . . , Ap. We shall prove t h a t cvovy A,- (i — 1, 2, . . . . /;) is a k-thin set. Let he a,\ , Oo • -.-, % - i t / l / . Wi5 shall d is t inguish th ree cases. 1. a,„ f(/\ p) for every m - 1, 2, . . . , k —- 1. I n th i s case we have: Oi O-j ; . . . ^7t.._! < d. F rom t h e construct ion of t h e set J ; it follows tha t Oi a-2 : . . . -f ^A-l £ - 4 * . 2. Let exact ly one of t h e n u m b e r s #/w be g rea te r t h a n d a n d the- o ther less or equal t o / ( / ' , />). We can assume t h a t j u s t Mi is g rea te r t han J a n d so Oo, '/:,, . . . . O/( i f: >ST/. Since tti > d >/(/<;, p), a\ is one of t h e numbers (4); hence <'((([) a\ d > f(k\ p) a n d from t h e construction, of t h e set A-t it follows, t h a t Oi d E »S'/. The set 8; is k-thin, hence we h a v e ( « i - - r f ) -h "a + . . . f flfc-i^^. We shall show t h a t a — tti -f- ttL> f . . . f a-jc-i $ Ai. W e shall prove indi-rectly. Assume t h a t a belongs t o A-,. Since a, > </, we can write O, — d > c(a); obviously c(a) = (ttl — (1) - f tt2 f . . . - f OL;-1 . From the construct ion of t h e set Af i t follows t h a t c(a) belongs t o S;,. This is a cont radic t ion . 3. Let a t least two of t he numbers am be grea ter t h a n d. Then we have «i ; "2 + . . . 4 "*-i > 2r/ > A/(k, T?) -f (X:— 2) (since /.* " 3) and therefore a\ -f- Oo I ••• f ^A- I ^ J « -The proof of t h e Theorem is comple ted , because t h e above considerat ions arc correct for a rb i t r a ry i - 1 ,2 , . . . , p. N o t e 3. The Theorem gives in fact also a m e t h o d of the direct spl i t t ing of the numbers k 2 I. 2. . . . . (/> - I) k 1 into p //-thin sets. We shall i l lus t ra te t h i s method on the case k 5, /; 3 I because of note 1 we have / ( 5 , 3) ^ »( r '3 - ! ) 9:3-The d:\ ision of t h e numbers i, 2, 03 in to th ree 5-thin sets is the following: J i J - 2, 3, ](>, 17, 18. 7b, 77, 78. 01 , 02, 03J .!- j4, 5 15, 70, 8 0 , . . . , 00} J s - !10, 2 0 , . . . , 75j-3 5 0 I l l We shall app ly the above resul ts to the solving of a known problem of the theory of g raphs . All considera t ions of pa r t I I I are direc t general isat ion of those of [1J. Le t g(k, p) denote the grea tes t na tura l n u m b e r such that all edges of a comple-te g raph of g(k, p) vertices can be coloured by p colours so t h a t there does not arise a complete subgraph of k vertices, all edges of which are coloured by the same colour. T h e existence of g(k, p) for a n y n a t u r a l k and p follows from the ar t icle [2J. Theorem 2. Let k V; 3 and p be natural numbers. We have: (5) 9(k, p) 7;.. f(k, P) 4- I-F r o of. Le t A\, A%, . . . , Av be such k-thin sets that each of the n u m b e r s 1 , 2 , . . . , / ( £ , p) belongs exact ly to one of them (existence of such sets follows from t h e definition of t h e n u m b e r / ( k , p)). Let G be a complete g raph with f(k, p) |- 1 vertices. Let us denote t h e m by Po, Pi, ..., P/(/.-,?/r • Colour all edges of g r aph G by t he colours d , Gt, •-, Gv in t h e following way: colour the edge in terconnect ing vert ices Pi and Pj by the colour G)N if and only if \i —j\ ? Am. Let us suppose t h a t all edges of a complete subgraph with I: vertices PUi P/a, P(k are in this colouring coloured by the >ame colour ( wo- ^ e ( ' a n slt]>l>ose t h a t (i\ — k), (k --- is), • -., (ifc-i — ik), (ii — ik) *:• A „ t n . Since AWo is a k-thin set, this is a contradic t ion, because (k -- iz) +• (22 — is) ( . . . -I- (ik I ik) G\ - ik)-The proof of Theorem 2 is complete . * * • The au tho r is thankful to Prof. T. Sala t for his very valuable notes regardim. this paper . RUFKKKNOKS j l j A b b o t t . 11. L.. M o s o r L., Sntn-free set* of inteaers. Acta a r i thn i . II (l.MW), W\)W— .{.•»;. | 2 j (» nHMLWood II. F . . (n l e a s o n A. M., (\)tnhinatorial relation* an</ e/ironxtt ><•</,•<! /J,s. Canad. J. Math . / (19r>">), 1 7. [3] R a d o R., Studien znr Komhinaforik, Math . Z. V) { I.)33). 424 4X0. 3 60 |4] S c l i u r L, Uber die Kongruenz xm | ym zm (mod p), Jahresber . Dtsch. Mat h.-Vor. :JÓ (1910), 1 14—117. |5j K n l o s I'., Sonie remarks on the theory of graphs, Bu l l . Amor. Ma th . Soc. 53 (1947), 292 299. Kecoived Oc tobc r 13, 1905. Katedra matemat iky Chemickoteclmologickej fakulty Slovenské] vysokej školy technickej, Bratislava ЗGI 

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Generalisation of a Number-Theoretical Result

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Institute of Mathematics AS CR, v. v. i.