Journal of Combinatorial Theory, Series A, May 1, 1996
We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X... more We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by c k } (ln n) 1Â(2k&1) } n (k&h)Â(2k&1) , where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0. This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d−1)... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d−1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.
Discrete Mathematics & Theoretical Computer Science, Nov 6, 1998
For an n-element set X and a proper coloring ∆: [X] k −→ {0, 1,. . .} where each color class is a... more For an n-element set X and a proper coloring ∆: [X] k −→ {0, 1,. . .} where each color class is a matching with cardinality bounded by u, we show that there exists a totally multicolored subset Y ⊆ X with |Y | ≥ max c1 • n k /u 1 2k−1 , c2 • n k /u 1 2k−1 • ln u/ √ n 1 2k−1 This bound is tight up to constant factors for u = ω(n 1/2+) for any > 0. Moreover, for fixed k, we give a polynomial time algorithm for finding such a set Y of guaranteed size.
We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X... more We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0 . This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices 1996 Academic Press, Inc. A typical anti-Ramsey type question, cf. [ESS75], [Ra75], is the following: Given is a graph or a hypergraph, where the edges are colored with certain restrictions on the coloring like, for example, edges of the same article no. 0049 209 0097-3165Â96 18.00
Bulletin of the Australian Mathematical Society, 1997
A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a... more A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g(r)(k) such that for every r-colouring of {1, 2, …, g(r)(k)} there is a monochromatic k-term member of 𝑐. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x1, …, xk} such that xi − xi−1 ≡ a (mod m) for i = 2, …, k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let be the collection of sequences that are either a(mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, has the 2-Ramsey property, and that the 2-Ramsey function , corresponding to k-term a(mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for . ...
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − ... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs. 1
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − ... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.
Journal of Combinatorial Theory, Series A, May 1, 1996
We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X... more We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by c k } (ln n) 1Â(2k&1) } n (k&h)Â(2k&1) , where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0. This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d−1)... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d−1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.
Discrete Mathematics & Theoretical Computer Science, Nov 6, 1998
For an n-element set X and a proper coloring ∆: [X] k −→ {0, 1,. . .} where each color class is a... more For an n-element set X and a proper coloring ∆: [X] k −→ {0, 1,. . .} where each color class is a matching with cardinality bounded by u, we show that there exists a totally multicolored subset Y ⊆ X with |Y | ≥ max c1 • n k /u 1 2k−1 , c2 • n k /u 1 2k−1 • ln u/ √ n 1 2k−1 This bound is tight up to constant factors for u = ω(n 1/2+) for any > 0. Moreover, for fixed k, we give a polynomial time algorithm for finding such a set Y of guaranteed size.
We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X... more We consider anti-Ramsey type problems for k-uniform hypergraphs. A subset Y of an n-element set X is totally multicolored, if the restriction of a coloring of k-element subsets of X to [Y ] k is a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinct k-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by where c k is a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X ] 2 with no totally multicolored subset of size 2.21n 1Â3 (ln n) 1Â3 , provided n n 0 . This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied with k=2 and h=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrices 1996 Academic Press, Inc. A typical anti-Ramsey type question, cf. [ESS75], [Ra75], is the following: Given is a graph or a hypergraph, where the edges are colored with certain restrictions on the coloring like, for example, edges of the same article no. 0049 209 0097-3165Â96 18.00
Bulletin of the Australian Mathematical Society, 1997
A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a... more A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g(r)(k) such that for every r-colouring of {1, 2, …, g(r)(k)} there is a monochromatic k-term member of 𝑐. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x1, …, xk} such that xi − xi−1 ≡ a (mod m) for i = 2, …, k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let be the collection of sequences that are either a(mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, has the 2-Ramsey property, and that the 2-Ramsey function , corresponding to k-term a(mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for . ...
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − ... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs. 1
Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − ... more Consider random regular graphs of order n and degree d = d(n) ≥ 3. Let g = g(n) ≥ 3 satisfy (d − 1) 2g−1 = o(n). Then the number of cycles of lengths up to g have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than g. A corresponding result is given for random regular bipartite graphs.
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