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Codes Cryptogr."],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A <jats:italic>partial<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$(n,k,t)_\\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mi>\u03bb<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:italic>-system<\/jats:italic> is a pair <jats:inline-formula><jats:alternatives><jats:tex-math>$$(X,{\\mathcal {B}})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>B<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where <jats:italic>X<\/jats:italic> is an <jats:italic>n<\/jats:italic>-set of <jats:italic>vertices<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {B}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>B<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a collection of <jats:italic>k<\/jats:italic>-subsets of <jats:italic>X<\/jats:italic> called <jats:italic>blocks<\/jats:italic> such that each <jats:italic>t<\/jats:italic>-set of vertices is a subset of at most <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> blocks. A <jats:italic>sequencing<\/jats:italic> of such a system is a labelling of its vertices with distinct elements of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{0,\\ldots ,n-1\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A sequencing is <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-<jats:italic>block avoiding<\/jats:italic> or, more briefly, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:italic>-good<\/jats:italic> if no block is contained in a set of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> vertices with consecutive labels. Here we give a short proof that, for fixed <jats:italic>k<\/jats:italic>, <jats:italic>t<\/jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bb<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, any partial <jats:inline-formula><jats:alternatives><jats:tex-math>$$(n,k,t)_\\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mi>\u03bb<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-system has an <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-good sequencing for some <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell =\\Theta (n^{1\/t})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>\u0398<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as <jats:italic>n<\/jats:italic> becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case <jats:inline-formula><jats:alternatives><jats:tex-math>$$k=t+1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> where results of Kostochka, Mubayi and Verstra\u00ebte show that the value of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cannot be increased beyond <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Theta ((n \\log n)^{1\/t})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u0398<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>log<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A special case of our result shows that every partial Steiner triple system (partial <jats:inline-formula><jats:alternatives><jats:tex-math>$$(n,3,2)_1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-system) has an <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-good sequencing for each positive integer <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\leqslant 0.0908\\,n^{1\/2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2a7d<\/mml:mo>\n                    <mml:mn>0.0908<\/mml:mn>\n                    <mml:mspace\/>\n                    <mml:msup>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-022-01085-5","type":"journal-article","created":{"date-parts":[[2022,7,26]],"date-time":"2022-07-26T12:10:25Z","timestamp":1658837425000},"page":"2375-2383","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Block avoiding point sequencings of partial Steiner systems"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9971-7148","authenticated-orcid":false,"given":"Daniel","family":"Horsley","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7963-9688","authenticated-orcid":false,"given":"Padraig","family":"\u00d3 Cath\u00e1in","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,7,26]]},"reference":[{"key":"1085_CR1","doi-asserted-by":"publisher","DOI":"10.1002\/9780470277331","volume-title":"The Probabilistic Method","author":"N Alon","year":"2008","unstructured":"Alon N., Spencer J.H.: The Probabilistic Method, 3rd edn Wiley, New York (2008).","edition":"3"},{"key":"1085_CR2","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1002\/jcd.21770","volume":"29","author":"SR Blackburn","year":"2021","unstructured":"Blackburn S.R., Etzion T.: Block-avoiding point sequencings. 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