Applied Mathematical Sciences, Vol. 6, 2012, no. 98, 4857 - 4861
Upper Bounds of Ramsey Numbers Decha Samana Department of Mathematics Faculty of Science, Chiang Mai University, Chiang Mai, Thailand
[email protected] Vites Longani Department of Mathematics Faculty of Science, Chiang Mai University, Chiang Mai, Thailand Abstract For positive integers s and t , the Ramsey number R(s, t) is the least positive integer n such that for every graph G of order n, either G contains Ks as a subgraph or G contains Kt as a subgraph. A widely known theorem, proved by Erd¨ os , state that s+t−2 R(s, t) ≤ s−1 In this paper, we improve the upper bounds for R(s, t) . That is, we find that s+t−2 s+t−4 − for s, t ≥ 5 R(s, t) ≤ s−1 s−2
Mathematics Subject Classification: 05C55, 05D10 Keywords: Ramsey numbers, upper bounds, graphs
1
Introduction
The problem of determining Ramsey numbers is known to be very difficult. The few known exact values and several bounds for different graphs are scattered among many technical papers[3]. For positive integers s and t , Ramsey number R(s, t) is the least positive integer n such that for every graph G of order n, either G contains Ks as a subgraph or G contains Kt as a subgraph. Some known R(s, t) are shown in the table[3]:
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Table 1. Known nontrivial values and some upper bounds for R(s, t) t s 3 4 5 6 7 8 9 10 11 3 6* 9* 14* 18* 23* 28* 36* 43 51 4 18* 25* 41 61 84 115 149 191 5 49 87 143 216 316 442 6 165 298 495 780 1171 7 540 1031 1713 2826 4553 8 1870 3583 6090 10630 9 6588 12677 22325 10 23556 * Exact Ramsey numbers For upper bounds, Theorem 1.1 and Theorem 1.2 are famous results shown by Erd¨os [2]. Better bounds for R(3, t) is expressed in Theorem 1.3, see [1] . Theorem 1.1 For every two positive s and t,
R(s, t) ≤
s+t−2 s−1
.
Theorem 1.2 For integer s ≥ 2 and t ≥ 2, R(s, t) ≤ R(s − 1, t) + R(s, t − 1). Theorem 1.3 Let c be a non-negative integer. Suppose that the Ramsey number R(3, t) satisfies t −c R(3, t) ≤ 2 for some t = t0 − 1 and t0 . Then the inequality holds for any t ≥ t0 − 1. Making use of the upper bounds in table 1. we are able to find c as follows. t0 − 1 8 9 c 0 0
10 11 12 13 2 4 7 9
14 15 13 17
Thus we have, for examples,
and
t R(3, t) ≤ 2 t − 17 R(3, t) ≤ 2
for t ≥ 8, for t ≥ 15.
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2
Main Results
In Theorem 2.1 and Theorem 2.2, we show that better results for R(4, t), when t ≥ 7 and R(5, t), when t ≥ 6, can be obtained. Theorem 2.1 For every integer t ≥ 7, R(4, t) ≤
t3 + 5t . 6
(1)
t3 + 5t = 63, so (1) holds 6 (t − 1)3 + 5(t − 1) for t ≥ 8. Consider if t = 7. Assume that R(4, t − 1) ≤ 6 R(4, t). By Theorem 1.2 and Theorem 1.3, we have Proof: For t = 7, from Table 1 R(4, 7) ≤ 61 while
R(4, t) ≤ R(3, t) + R(4, t − 1)
t3 − 3t2 + 8t − 6 t + ≤ 2 6 2 3 2 t − t t − 3t + 8t − 6 + = 2 6 t3 + 5t − 6 = 6 t3 + 5t ≤ 6 t3 + 5t for t ≥ 7. R(4, t) ≤ 6
Therefore
Upper bounds R(4, t), for 7 ≤ t ≤ 15 are shown in the following table. The second, and the third lines in the table are upper bounds obtained by using Theorem 1.1 and Theorem 2.1 respectively. R(4, t) 7 8 9 10 11 12 13 14 15 Theorem 1.1 84 120 165 220 286 364 455 560 680 Theorem 2.1 63 92 129 175 231 298 377 469 575 Theorem 2.2 For every integer t ≥ 6, R(5, t) ≤
t4 + 2t3 + 11t2 + 10t . 24
Proof: For t = 6, from Table 1 R(5, 6) ≤ 87 while
t4 + 2t3 + 11t2 + 10t = 91, so (2) holds if t = 6. 24
(2)
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Assume that R(5, t − 1) ≤
(t − 1)4 + 2(t − 1)3 + 11(t − 1)2 + 10(t − 1) for t ≥ 7. 24
Consider R(5, t). By Theorem 1.2 and Theorem 2.1, we have R(5, t) ≤ R(4, t) + R(5, t − 1) t3 + 5t t4 − 2t3 + 11t2 − 10t + 6 24 t4 + 2t3 + 11t2 + 10t = 24 4 3 t + 2t + 11t2 + 10t R(5, t) ≤ for t ≥ 6. 24 =
Therefore
Upper bounds R(5, t), for 6 ≤ t ≤ 1 are shown in the following table. The second, and the third lines in the table are upper bounds obtained by using Theorem 1.1 and Theorem 2.2 respectively. R(5, t) 6 7 8 9 10 11 Theorem 1.1 126 210 330 495 715 1001 Theorem 2.2 91 154 246 375 550 781
12 1365 1079
13 14 1820 2380 1456 1925
For Theorem 2.1 and Theorem 2.2, we can rewrite t3 + 5t = R(4, t) ≤ 6
and
t+2 t − 3 2 4+t−2 4+t−4 = − for t ≥ 7, 4−1 4−2
t4 + 2t3 + 11t2 + 10t t+3 t+1 R(5, t) ≤ − = 4 3 24 5+t−2 5+t−4 = − for t ≥ 6. 5−1 5−2
For general results, we have Theorem 2.3. Theorem 2.3 For all integer s, t ≥ 5, the Ramsey number
R(s, t) ≤
s+t−2 s−1
−
s+t−4 s−2
.
(3)
Proof: We proceed by induction on k = s + t. We can verify that (3) holds when k = 10. For examples, from Table 1,
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and
5+5−2 5+5−4 − = 53 R(5, 5) ≤ 49 while 5−1 5−2 4+6−2 4+6−4 R(4, 6) ≤ 41 while − = 41. 4−1 4−2
Consider when s ≥ 5, t ≥ 5 with k ≥ 11. Suppose (3) hold when s + t < k, i.e.
R(s , t ) ≤
s + t − 2 s − 1
−
s + t − 4 s − 2
.
Let s and t be such that s + t = k. By the inductive hypothesis, Theorem 2.1, and Theorem 2.2, it follows that
s+t−3 s+t−5 R(s − 1, t) ≤ − s−2 s−3 s+t−3 s+t−5 and R(s, t − 1) ≤ − . s−1 s−2 By Theorem 1.2, we have R(s, t) ≤ R(s − 1, t) +R(s,t − 1) s+t−3 s+t−5 s+t−3 s+t−5 ≤ − + − s−2 s−3 s−1 s−2 s+t−3 s+t−3 s+t−5 s+t−5 = + − − s−2 s−1 s−3 s−2 s+t−2 s+t−4 = − . s−1 s−2 s+t−2 s+t−4 Hence R(s, t) ≤ − . s−1 s−2 s+t−2 s+t−4 − for all s, t ≥ 5. Therefore R(s, t) ≤ s−1 s−2
References [1] C. Nara and S. Tachibana, A note on upper bounds for some Ramsey numbers, Discrete Math., 45(1983) 323-326. [2] P. Erdos and G. Szekers, A combinatorial problem in geometry, Compositio Math., 2 (1935) 463-470. [3] S.P.Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey 1, revision11, August 2006, http://www.combinatorics.org. Received: February, 2012
