Znám, Štefan

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

Matematický časopisŠtefan ZnámOn k-Thin Sets and n-Extensive GraphsMatematický časopis, Vol. 17 (1967), No. 4, 297--307Persistent URL: http://dml.cz/dmlcz/127008Terms of use:© Mathematical Institute of the Slovak Academy of Sciences, 1967Institute 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.czMatematický Časopis 17 (1967). No. 4 ON k-THIN SETS AND n-EXTENSIVE GRAPHS ŠTEFAN ZNÁM, Bratislava This article is a sequel to paper [7]. We deal here with the generalised form of a well-known problem from the theory of numbers (originated in 1916 by I. Schur, see [6]); further we give some applications of results in the graph theory (see [4]). Let k, n and p be natural numbers, with k ^ 3. Definition 1. We say that the set M is a k4hin set if from the condition a\, a%, ..., aic-i e M it follows that a± + a2 + ... +• ak-x £ M (the numbers a% need not be different). Definition 2. The greatest natural number N for which there exist p disjoint k-thin sets Si, 82, ..., Sp such that {n,n + \,...N}=\JSi9 *=i will be denoted by f(k9 n9 p)(*). Remark. The/(&, \9p) is identical to f(k9p) introduced in paper [7]. In the present paper we shall determine the value, resp. the lower estimation off(k, n,p). Our main result is Theorem I. For arbitrary natural k ^ 3, n and p w& have (1) /(*, n, p) £ kf(k, n,p-\) + (k-n-\). t1) From results of [5] the existence of f(k, n, p) for an arbitrary natural k ^ 3, n and p. follows. 297 Remark. The special fcase of this Theorem (case n = 1) is proved in paper [7]. The proof of Theorem I is completely analogical to the proof of its special case and therefore we shall not demonstrate it here. Corollary 1. Let u and v be natural numbers, with v ^ u. We have (2) f(k, n, v) ^ k*-uf(k, n, u) -\ (b>~u — 1). k — 1 Proof of Corollary 1. If we apply the inequality (1) to the number f(k,n,v), we have: f(k, n, v) ^ kf(k, n, v - 1) + (k - n — 1) ^ k*f(k, n, v — 2) + + k(k — n — 1) + (k — n — 1) ^ ... £ &»-*/(&, r^9 u) + + (k — n — 1) (fc»-«-i + i*-«-a + ... + fc + 1) == = k*-uf(k, n, u) H (£*-" — 1). AJ —- 1 Corollary 2. (3) /(*, n, p) £ - ^ - (kP - l)n + (n - 1). k — 1 Corollary 2 is a special case of Corollary 1 (case of u = 1; obviously /(&, n, 1) = (k — l)n — 1 for arbitrary k and n), but we mention it separately due to its great inportance: it gives a lower estimation oi f(k,n,p). The estimation (3) is not the best possible and for the case of n = 1, k = 3 it was already improved (see [1]). For an arbitrary p ^ 4 it is true that 89 . 3*>-4 — 1 f(3,l,p) £ > which is obviously better than the estimation following from (3): 3 * - - 1 f(3,l,p) 2 s — — . Again, in the case of p = 2 we have in (3) an equation for an arbitrary n and k ^ 3. We state it in the form of a Theorem: Theorem II. For an arbitrary n and k ^ 3 we have k 2 f(k, n, 2) = (jfca -- l)n + (n — 1) = (B - & - l)n - 1. & — 1 29$ • r Remark . Hence in the case of p v£ 2 our problem is solved completely* since we found the exact value oif(k, n, 2). Proof of Theorem I I . From (3) it follows that f(k, n, 2) ^ (k2 — k—\)n—\. To finish our proof we must show that f(k,n,2) <, (k2-k- \)n- 1. Hence it is suffucient to show that the numbers ' , x (4) n,n+ \,...,(k2-k- \)n cannot be divided into two &-thin sets for any n and k ^ 3. We shall use the methods used for the proof of analogical assumptions in paper [8]. We shall prove indirectly. Let us suppose that there exists a division of the numbers (4) into two &-thin sets. Let us denote them A and B. Without loss of generality we can suppose that neA. Since the sum of k —r 1 elements of A cannot belong to A, the number (k — \)n belongs to the set JS. From analogical considerations it follows that the number (k — \)2n = (k2 — 2k+ \)n v\ belongs to the set A (this number is smaller than (k2 — k — \)n). We can write (k2 — k — \)n = (k — 2) . n + 1 . (k2 - 2k + \)n. The numbers (k2 — 2k + \)n and n are from the set A, hence the number (k2 — k — \)n belongs to the set B. Now we shall distinguish two cases: a) Let kn e A. We have: n, kn, (k2 — 2k + \)n e A, where (k2 — 2k + \)n = 1 . n + (k — 2) . kn (kn is smaller than (k2 — k — \)n, since k ^ 3). It is a contradiction, because A is a it-thin set. b) Let kn e B. We have (k — \)n, kn, (k2 — k —- \)n eB, where (k2 - k - \)n = 1 . (k — \)n + (k - 2). kn. I t is a contradiction, because B is a i-thin set. From a) and b) it follows that the number kn cannot belong to any of the sets A and B; hence the numbers (4) cannot be divided into two fc-thin sets in any way; q. e. d. Remark. Our method — so simple for the case of p = 2 — is already very .299 complicated for the case of p == 3. To prove that the numbers 1, 2, ,„., 14 cannot be divided into three 3-thin sets in any way, we must distinguish 17 cases (we have here k = 3, n == 1, p = 3). In the cases p ^ 4 it is advisable to use computers. * * * We give now the second proof of the relation (3) (independent of Theorem I), which gives a good method for the direct division of the numbers (5) n, n + 1, ..., ^— (kP - l)n + (n - 1) into p &-thin sets. Second proof of (3). First we prove relation (3) for the case of n = 1, i. e. we prove that the numbers fc — 2 (6) 1,2, ..., ; ( f c > - l ) k — 1 can be divided into p &-thin sets. Let us form from the numbers (6) the following sets: Fx = {x : x = 1, 2, ..., (k — 2) (mod (k — 2)k)}, F2 = {x : x = (k - 1), k, ..., (k - 2)k (mod (k - 2)£2)}, ( *—2 • 1 Fm = \x:x = * (&™-i - 1) + 1, ..., (k — 2)i»-i (mod (fe — 2)£»») , ľ . = Jfc-2 x : x = (kP-1 — 1) + 1,..., (k — гjfcî'-1 (mod {k — 2)kv) ÍC — 1 Now we prove that a) all Fm are .fe-thin sets, b) every number from (6) belongs to at least one of the sets Fm. a) liet Xi, x2> „*,xjc~.i eFm (where m is an arbitrary of the numbers 1, 2, . . . ,p). From the construction of Fm it follows that we can find such numbers j/i, 2/2, ..-* 2/*-i that we have xi = yi (mod (k -- 2)km), x2 = y2 (mod (k — 2)km), ..., Xk-i = 2/k-i (mod (k — 2)km), ^where £ 2 (7) 7 (&™-1 - 1) + 1 S yu 2/2, ..., y*-i ^(k- 2)km~K k — 1 . „ .-306 From (7) it follows:' *—i (8) 2 'Sfi -S (* - 1) (*> - 2)t»-i < (i - 2)i*, ^ - i *—i (9) 2 VJ^ (k - 2 ) (fcw-1 -^ i) + (* - 1 ) -= (* - 2)^-1 + 1 . * - i Because of (8) and (9) we have: *—I (fc - 2)fc™-i < 2 y; < (h - 2)^w-. i = i &—1 From the last inequality it follows that the number 2 yj cannot be congruent ' i - i with any of the numbers of Fm (mod (k -— 2)&w). The same holds for the number k—1 Jfc—1 k—1 • * — 1 2 #/, since 2 ^ — 2 W (m°d (& — 2)fcm). Hence 2 ^ ?=i i = i i = i ^ - i cannot be equal to any of the numbers of Fm and Fm is a fc-thin set. Since m was an arbitrary of the numbers 1, 2, ..., p, the proof of part a) is finished. b) We have to prove that each of the numbers of (6) belongs at least to one of the sets Fm. We shall prove it by induction with respect to p. It is very easy to verify that the assertion is valid for p = 1. Let p > 1. Let us suppose that the assertion is valid for p — 1 (i. e. that the numbers 1,2, . . . , - - = ^ ( ^ - 1 - 1 ) k —• 1 belong to the sets F\, Fz,..., Fp-i). We shall prove that the assertion is valid for p, too. The numbers _---_ (fcp-l _ 1) + 1, - JL- (fcp-l - 1) + 2, ..., (* - 2 ) * P - I k — 1 k — 1 obviously belong to the set FP> We must prove yet that each of the numbers Jfc_-2 *•< (10) (k - 2)fc>-i + 1, (k - 2)fc2>-i + 2, ..., (kP - 1), . k — 1 / belongs at least to one of the sets Fm. Since (fc - 2)fc*-i + ; (kv-i - 1) =* (Jfcp - 1), ' * — 1 r̂  fc — 1 , %, 3Qi every of the numbers (10) can be written in the form (k — tykv-1 +1Y, where 1 S T g —— {fc*"1 — 1). k —- 1 Because of the inductional assumption every such Y lies at least in one of the sets Fi,F2, ...,Fp-i. The same is valid for the numbers (10), since they are congruent with the related Y (mod (k — 2)&->T1}, hence also (mod (k — 2)k*), where 1 ^ s ^ p — 1. A number from (10) belongs therefore into the same set as the Y related to it. " i -The sets Fm are not disjoint, but we can easy get from them a system of disjoint sets. The proof of (3) for n = 1 is finished. : Now we prove (3) for an arbitrary natural n > 1, i. e. we prove that the numbers (5) can be divided intQ p A>thin sets. Let us divide the numbers (5) into 7i-tuples in the following way: ai = [n, n + 1, ..., 2n —- 1}, a2 = {2n, 2n + 1, ..., Sn — 1}, at = {in, in + 1, ..., (in + n — 1)}, i [k — 2 - . , k — 2 ,-. ) ak^ = (kP ±- \)n,::., (kv - l)n + (n - 1) . R^1 1 lfc-lV« ' ^ J f c - l V ' . . : V M . . • ' 7 ' K:v ' j;v.-.;. . . . . i rLet us form from the numbers (5) the, aets Gi,G2, ...,GP in the folloMdng way: we put the whole 7i-tuple ai iii the set Gm if and only if i belongs to the set Fm (where Fm are the sets introduced above). Every number from (5) belongs exactly to one w-tuple, every n-tuple belongs to at least one set Gm (since each of the numbers in (6) belongs to at least one Fm), hence each of the numbers in (5) belongs to at least one of the sets Gm. We must prove yet that Gm are &-thin sets. Let xi, x2, ...,X]c-ieGm (m is an arbitrary of numbers 1,2, ...,p). Let us denote (11) xi + x2+ ... + xk-i = x0. It is well-known that every number xt (i t*-= 0, 1, . . . , k — 1) can be written in the form: (12) Xi = rin+qi, where ri and qt are not negative integers, and (13) qi^n—1. If we put (12) in (11) we shall have 302 k—1 Jfc—1 ' , , r (14) n 2 r « + 2 qi = r0n + q0. , ,-.;• i = l i = l Jfc—1 According to (13) 2 ?* < (^ — 1)%> hence we can write i = l . k—l (i5) 2 # = ( f c - j > + ?> i = i . . .: =• : *• where . . • . . - -(16) 2 ^ j £ k, 0 ^ g ^ n — 1. \^ From (14) and (15) we have the following equation k—1 (17) n 2 n + (k -r.j)n + q = r0n + q0. i = l From (13), (16) and (17) it follows that qo — #, hence ]• *—I 2 n + (k—j) = r0. i = l According to the assumption the numbers xx,X2, ..., a;*—I belong to the set Gm% therefore from the construction of w-tuples and the sets Gm it follow^ that the numbers n , n>..., n-i belong to the set Fm. Further from the construction of the sets Fm the existence of such numbers U(i = 1,2, ..., k — 1) follows that n = U ( m o d (& — 2)fc™), where *—I (18) (Jfc - 1) + (k - 2) (Jfc»-i - 1) ^ 2 ** ̂  (fc - l) C° - 2 ) ^ ^ i = l (see (7), (8) and (9)). From (16) and (18) we get the inequalities: 2 > + (fe - i ) ^ (fc - 2 ) t ^ " 1 - i) + (* - 1 ) + (* - i) X * - 2)t«-i, i « - l Jfc—1 2 *t + (Jfc - j) ^ (fc - 2) (Jfc - l)Jfc»-i + (k-j) £ (Jfc - 2)fc*»< i = l Jfc—1 Hence 2 k + (h — j) $Fm, thus the number i - l jfc—I * — I r0 = 2 n + (k - j) = ' 2 ' * + (fc - J ) ( m o d (& - 2) i w) i = i * = i ЗФЗ is not from Fm either. But from this it follows that x0 = ron + qo $ Gn The proof of (3) is completed. n In this part we shall show an application of the above results by the solving of a well-known problem from the graph theory* Definition 3. Let n and N be arbitrary natural numbers. We shall say that a graph G of N vertices is an n-extensive graph if we can denote all vertices of G withnumbersO, 1, ...,N —- 1 so that two vertices Pi and Pj(i,j = 0, 1, ...,N— 1) are connected by an edge if and only \i — j \ ^ n. Remark. Obviously every complete graph is an 1-extensive graph. Definition 4. Let the natural numbers n, p and ki ^ 2 (i = 1, 2, ...,p) be given. We shall denote by g(n,p; &i, &2, ..., kp) the greatest natural number K for which all edges of an arbitrary n-extensive graph of K vertices can be coloured by p colours so that there does not arise any complete subgraph of ki vertices, all edges of which are coloured by the same colour Ct(i = 1, 2, ...,p)(2). Definition 5. A complete subgraph, all edges of which are coluored by the same colour (Ci) will be called monochromatic (C(-chromatic). Papers [4] and [7] deal with the case of n -=. 1 (i. e. with the oase of the complete graph). The results of our paper give a generalisation of the results of [4] and [7]. We shall determine the lower and the upper estimation of the function g(n,p;klyk2, ...,kp). Theorem III. For an arbitrary natural n,p and ki ^ 3 (i = 1, 2, ...,p) we have (19) g(n,p;k1,k2, ...,kP) ^ v ^ 2 9(n>Plh> •••, fci-ii h — l,k{+1, ...,kp) + n. i = l Proof, Let G be an ^-extensive graph of p JV= 2 g(n>p;fa, ...,-W-i,ft* — 1,fo+i, ...,fcp) + n+ 1 i = l (2) The existence of the number K = g(n, p; ki, &2, . . . , kp) for n = 1 follows from the article [4]; for n > 1 we shall prove it in our paper. 304 vertices. We shall prove indirectly. Let us suppose that we find such a colouring of all edges of G by p colours that there does not arise any O^-chromatiq complete subgraph of ki vertices (i = 1, 2, . . . , p ) . Let us denote the vertices of G with numbers 0, 1, . . . , N — 1 in such a way that two vertices Pi and Pj are connected by an edge if and only if \i —- j \ ^ n (it is obviously possible, because G is an w-extensive graph of N vertices). The vertex denoted by 0 denote by Vo. There exist exactly n — 1 vertices which are not connected with Vo by an edge. Let Ti denote the set of this vertices of G which are connected with Vo by an edge of colour Ci. Let the number of elements of Tt be mi. Then we have: 1 + 1 rrn + (n - 1) =-- N. *=i From this it follows that we cannot have for every i(= 1, 2, . . . , p) inequality rrn ^ g(n,p;ki, ...,kt-i,ki — l,ki+i, ...,kp) but there exists at least one i for which m >g(n,p;ki, . . . ,4j_i , fo — l,ki+i, ...,kp). Whence it follows that in Ti either there exists a O«-chromatic(s 4= i) complete subgraph of k8 vertices, or there exists a C*-chromatic complete subgraph of ki — 1 vertices. If we give to the later the vertex Vo (which is connected with all vertices of Ti by an edge of colour d) we shall have a Oi-chromatic complete subgraph of ki vertices. It is a contradiction and the proof of the Theorem is finished. Remark 1. A special case of Theorem III (n = 1, i. e. the case of complete graphs) is proved in paper [4], the methods of which are used in our paper. Remark 2. Obviously g(n,p; ki, ..., 4f_i, 2, ki+i, ...,kp) = g(n,p — 1; ki, . . . , ki-i, ki+i, ... kp), therefore by (19) it can be proved by induction (with respect to p) that the function g(n, p; ki, ..., kp) is finite for an arbitrary n, p and ki (i = 1, 2, . . . , p). Remark 3. We can state further: by analogical considerations as in the case of n = 1 (see [4]) we can prove the inequality (20) g(n,p;h,...,kp) < ( ^ ' " ^ ^ + *<* + l){k> + " + kp) kil ... kp\ This upper estimation of the function g(n, p; ki, ... kp) is very rough and probably can be essentialy improved. For the case of n = l a better estimation is shown in paper [4]. 305 Now we shall deal with the lower estimation of the function g(n, p\ki,...,kp). Determining lower estimations we shall use the results of part I of our article. A connection between the problem of colouring the edges of a graph and the problem of division of numbers into &-thin sets was shown first in paper [1]. Further we shall consider only the case &i = fc2 == ... = kp = k. For the sake of simplification we introduce the notation g(n,p; &,..., k) = g(k, n, p). Hence g(k, n, p) is the greatest of such natural numbers for which all edges 6f any w-extensive graph of g(k, n, p) vertices can be coloured by p colours so that there does not arise any monochromatic complete subgraph of k vertices. Theorem IV. For an arbitrary natural k(^ 3), n and p we have (21) g(k,n,p) ̂ f(k,n,p)+ 1. Proof. Let G be an arbitrary n-extensive graph of N =f(k,n,p) + 1 vertices. Let us form p such &-thin sets Ii,l2, ..., Ip that each of the numbers n, n + 1, ...,f(k, n,p) belongs exactly to one of them (existence of such sets follows from the definition of/(&, n,p)). Let us denote the vertices of G with the numbers 0, 1, ..., N — 1 so that two vertices Pi and Pj are joined by an edge if an only if \i — j \ ^ n (the possibility of such notation follows from the assumption that G is an ^-extensive graph of N vertices). The edge joining the vertices Pr and P8 is coloured by the colour Cm(m = 1, 2, ...,p) if and only if \s — r\ e Im. We shall show that this colouring fullfils the demands, i. e. there does not arise any monochromatic complete subgraph of k vertices (Obviously each edge of G ^coloured exactly by one colour). We shall prove indirectly. Let us suppose that by this colouring there arises a Cj-chromatic (i = 1, 2, ...,p) complete subgraph with the vertices P. P. P. We can suppose that i i > i 2 > . . * > i * . G is an n-extensive graph, hence we have: n ^ ii — i2, n ^ i2 — i3, ..., n ^ ijt_i — ik, n ^ ii — i*. According to the assumption all edges of this complete subgraph are coloured by the same colour Ct and so we have ii — i2 GIt , i2 — i$eli, ..., i*-i — ik e I i , ii — ik e-f<. Obviously the following is valid (ii — i2) + (i2 — i3) + .. . + (ijt-i — ik) = (ii — ik). 306 But this is a contradiction because It is a &-thin set. The proof of the Theorem is completed. R e m a r k 1. Special cases of (21) are proved in the papers [1] and [7]. R e m a r k 2. I t is easy to verify the following assertion: Let G be a subgraph of an Wfextensive graph G' of N = f(k, n, p) + 1 vertices. All edges of G can be coloured by p colours so that there does not arise any monochromatic complete subgraph of k vertices. R e m a r k 3. From (3) and (21) we have: Jfc — 2 g(k, n,p) ^ (kP — \)n + n. k — 1 I t is a good lower estimation only for the case of a small k (for the case of a great k see [2]). R e m a r k 4. From (20) and (21) we have the inequality (pk)\ f(k,n,p) tz—- + n(p + l)P*-l, (k\)P Which gives an upper estimation oif(k, n,p). REFERENCES [1] A b o t t H. L., Moser L., Sum-freesetsof integers, Actaarithm. 11 (1966),393—396. [2] Erdős P., Some remarhs on the theory of graphs, Bull. Ameг. Math. Soc. 53 (1947), 292—299. [3] Erdős P., Graph theory and proЪalitity II., Canad. J. Math. 13 (1961), 346—352. [4] Greenwood R. E., G l e a s s o n A. M.9Combinatoricalrelationsandcћromatic graphs, Canad. J. Math. 7 (1955), 1—7. [5] R a d o R., Studien zur KomЫnatorih, Math. Z. 36 (1933), 424—480. [6] Schur I., Űber die Kongruenz xm + Уm ----- zm (mod p)9 Jahг sber. Dtsch. Math.-Ver. 25 (1916), 114—117. [7] Z n á m Š., Generalisation of a numЪer theoretical result, Mat.-fyz. časop. 16 (1966)* 357—361. * [8] Znám Š., Megjegyzéseh Turán Pál egy puЫihálaťlan eredményéhez, Mat. lapok 16 (1963), 307—310. Received May 24, 1966. Katedra matematihy ChemichotecҺnologichej fahulty Slovenshej vysohej šholy technichef9 Bratislava 307 

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On $k$-Thin Sets and $n$-Extensive Graphs

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On $k$ -Thin Sets and $n$ -Extensive Graphs

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On kkk-Thin Sets and nnn-Extensive Graphs

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On $k$ -Thin Sets and $n$ -Extensive Graphs