Ex(n;t)

Values shown in bold are known exact values. Other values are lower bounds. If two values are shown then they are the lower and upper bounds.

Table of some known values for ex(n;t) for n=1,2,...,50 and t=3,4,...,13.

n/t	3	4	5	6	7	8	9	10	11	12	13
1	0	0	0	0	0	0	0	0	0	0	0
2	1	1	1	1	1	1	1	1	1	1	1
3	2	2	2	2	2	2	2	2	2	2	2
4	4	3	3	3	3	3	3	3	3	3	3
5	6	5	4	4	4	4	4	4	4	4	4
6	9	6	6	5	5	5	5	5	5	5	5
7	12	8	7	7	6	6	6	6	6	6	6
8	16	10	8	8	8	7	7	7	7	7	7
9	20	12	10	9	9	9	8	8	8	8	8
10	25	15	12	11	10	10	10	9	9	9	9
11	30	16	14	12	12	11	11	11	10	10	10
12	36	18	16	14	13	12	12	12	12	11	11
13	42	21	18	15	14	14	13	13	13	13	12
14	49	23	21	17	16	15	15	14	14	14	14
15	56	26	22	18	18	16	16	15	15	15	15
16	64	28	24	20	19	18	17	17	16	16	16
17	72	31	26	22	20	19	18	18	18	17	17
18	81	34	29	23	22	21	20	19	19	18	18
19	90	38	31	25	24	22	21	20	20	20	19
20	100	41	34	27	25	23	23	22	21	21	21
21	110	44	36	29	27	25	24	23	22	22	22
22	121	47	39	31	29	27	25	24	24	23	23
23	132	50	42	33	30	28	27	26	25	24	24
24	144	54	45	36	32	29	28	27	27	26	25
25	156	57	48	37	34	30	30	28	28	27	26
26	169	61	52	39	36	32	31	30	29	28	28
27	182	65	53	41	38	33-34	32	31	30	30	29
28	196	68	56	43	40	34-36	34	33	32	31	30
29	210	72	58	45	42	36-38	36	34	33	32	32
30	225	76	61	47	45	37-41	37-38		34	33	33
31	240	80	64	49	46	39-43	38-40		36	34	34
32	256	85	67	51	47	40-45	40-42		37	36	35
33	272	87	70	53-54	49	41-47	42-44			37	36
34	289	90	74	55-56	51	43-50	43-46			39	38
35	306	94	77	57-61	53	45-53	44-48			40	39
36	324	99	81	59-63	55	46-55	46-50				40
37	342	104	84	61-65	56-58	47	48-52				42
38	361	109	88	62-68	58-60	49	49-54				43
39	380	114	92	64-70	60-63	51	50-56
40	400	120	96	67-73	62-65	53	52-58
41	420	124	100	69-75	64-67	54	54-60
42	441	129	105	71-77	65-69	56	55-62
43	462	134	106-108	73-80	67-71	58	57-64
44	484	139	108-112	75-82	69-73	60	59-66
45	506	144	110-116	77-85	71-76	61	60-68
46	529	150	114-119	80-87	73-78	63	61-70
47	552	156	118-123	82-90	75-80	65	63-72
48	576	162	122-127	84-92	77-82	67	64-74
49	600	168	125-131	87-95	79-84	69	66-76
50	625	175	130-135		81-87	70	68-78

The following table is the key to the colors in the table presented above:

Color	References
*	Folklore.
*	Mantel's theorem for t=3 <ref>Mantel's theorem</ref>
*	Garnick, Kwong and Lazebnik<ref>D. K. Garnick and Y. H. H. Kwong, F. Lazebnik, Extremal graphs without three-cycles or four-cycles, J. Graph Theory, 17 (1993), 633--645</ref>.
*	Garnick and Nieuwejaar<ref>D. K. Garnick and N.A. Nieuwejaar, Non-isomorphic extremal graphs without three-cycles or four-cycles, JCMCC, 12 (1992), 33--56</ref>.
*	E. Abajo and A. Diánez <ref>E. Abajo, A. Diánez, Size of Graphs with High Girth, Electronic Notes in Discrete Mathematics 29 (2007) 179--183</ref>
*	E. Abajo and A. Diánez <ref>E. Abajo, A. Diánez, Exact values of ex(v;{C3,C4,... , Cn}), Discrete Applied Mathematics 158 (2010) 1869--1878</ref>
*	Tang et al. <ref>J. Tang, Y. Lin, M. Miller, C. Balbuena, Construction of EX graphs, Special Issue of International Journal of Computer Mathematics (2009)</ref>
*	Delorme et al. <ref>C. Delorme, E. Flandrin, Y. Lin, M. Miller and J. Ryan, On Extremal Graphs with Bounded Girth Electronic Notes in Discrete Mathematics, Volume 34, 1 August 2009, Pages 653-657</ref>
*	Delorme et al. <ref>C. Delorme, E. Flandrin, Y. Lin, M. Miller, K. Marshall and J. Ryan, New exact values of ex(n;6), preprint</ref>
*	Marshall et al. <ref>K. Marshall, M. Miller and J. Ryan, Extremal Graphs without Cycles of length 8 or less, preprint</ref>
*	K. Marshall and Y. Lin <ref>K. Marshall and Y. Lin, Extremal {C3,C4,.. ,C9}-free graphs, submitted IWOCA 2011t</ref>

References

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